That's pretty much it:
Is it possible to create a relativistic space probe going at least 0.1c with present day technology?
Present day meaning as of April 2020. If it is actually possible, how could it be done, what would it cost, what could be gained from doing it, and why has no one done it before?
No.
10% of the speed of light is about 30,000,000 m/s. Our fastest space probe to date, New Horizons, left Earth at less than 1/1000 of that speed. With a large propellant tank and a high-efficiency ion thruster we could reach something like 300,000 m/s, about a tenth of 1% of the speed of light. Due to the exponential nature of the rocket equation, reaching higher speeds requires exponentially larger amounts of propellant.
I'm showing the calculations for Russell Borogove's excellent answer.
You've asked to accelerate an object to 0.1 times the speed of light. Mathematically, $$\left( \frac{\Delta v}{c} \right) = 0.1$$
The exhaust velocity $v_e$ of an ion thruster is 20-50 km/s. Let's pick $v_e=30,000 \text{ m/s}$, thus $$\left( \frac{v_e}{c} \right) = 10^{-4}$$
And let's say the mass of our payload is 1 kilogram: $$m_1 = 1 \text{ kg}$$
The special relativity form of the rocket equation is $$\Delta v = c \tanh\left(\frac {v_e}{c} \ln \frac{m_0}{m_1} \right)$$
Solving for the initial (fueled) rocket mass $m_0$, $$m_0 = m_1 e^{\left(\frac {c}{v_e}\right)\tanh^{-1}\left(\frac{\Delta v}{c}\right)} = 1 \text{ kg } e^{10^4 \tanh^{-1}0.1} \approx 10^{435} \text{ kg}$$
The mass of the observable universe is estimated to be only 1.5x10$^{53}$ kg.
I would say NO.
Breakthrough Starshot claims to be capable of attaining 0.15c to 0.2c. But, the concept is based on a swarm of tiny probes (centimeter scale). They would be propelled by a "ground-based" laser; no on-board propellants circumvents the tyranny of the rocket equation. Breakthrough Starshot depends on a number of technologies not yet available or sufficiently advanced to meet required parameters. For relativistic spaceflight, it seems like the closest thing to achievable today, if you measure "close" in decades.
The fastest manmade object is a "hubcap" that was used to cover a nuclear blast testing site, which was clocked at 125,000 miles per hour. With a spacecraft designed to use nuclear bombs for propulsion, it has been suggested that it may be possible to build a spacecraft that can reach .1c with modern technologies, though doing so would require solving some engineering challenges first. NASA's Project Longshot, for instance, was calculated as having a top speed of .045c on a trip to Alpha Centauri, and would presumably be able to reach a speed of approximately double that if they used all their fuel without leaving any for deceleration.
Given that rockets are clearly not cut for this, I find it rather weird that, despite a couple mentions in the comments, Breakthrough Starshot isn't getting more discussion here even though it was literally the first thing that came to my mind when I saw this question up. And that leads one to naturally consider the state of the art of beam propulsion, for that is what is relevant here.
Beamed propulsion, of course, circumvents the rocket equation by leaving the fuel on the ground, so that the "have to lift fuel to lift fuel to lift fuel to ..." business that creates the exponential problem with rockets no longer applies.
Now, the simplest method to do beamed propulsion, perhaps, is a laser and, indeed, BTSS aimed to use exactly such. Given that BTSS is not expected to produce results for some 50 years or more (iirc), then I'd say this isn't "present day" by your definition but, given that posts have at least examined the feasibility of using existing rockets, I find it is only fair to try a similar at least cursory analysis of the existing possibilities for laser beam propulsion.
Beam propulsion, of course, works on the principle that light carries momentum as well as energy and, so, if suitably directed at a craft, can create a force (transfer of momentum) upon it. The relevant equation is Einstein's
$$p = \frac{E}{c}$$
where $E$ is the energy in the beam of light. If the spacecraft is an ideal reflector, it will manage to acquire twice this much because the beam is reflected back, and that back-reflection must be balanced by an extra forward momentum equal to the whole original beam thanks to the conservation of momentum.
Note, of course, that there is the factor $c$ in the denominator, which, in human scale units, is crazy big: as a result, even a modest energy will only produce a little extra momentum and, hence, only minimal acceleration of a spacecraft. In particular, using $p = \gamma mv$ for a general relativistic spacecraft, we see the energy required to accelerate to speed $v$ is
$$E_\mathrm{accel} = \frac{\gamma mc v}{2}$$
for the ideal-reflection case. Likewise, if we are alloted a certain amount of energy and want a certain goal speed, we can figure the maximum mass:
$$m_\mathrm{max} = \frac{2E}{c\gamma v}$$
So how much laser energy can we reasonably muster? Well, there apparently was one laser from as far back as the 1980s called "MIRACL", which was a chemical gasdynamic laser meaning that instead of electric power it was fueled directly by a special chemical fuel and obtained a peak power exceeding 1 MW and 70 s firing time, which means you can have 70 MJ to play with.
Since it has been built, it could be again and, perhaps better now. Thus I'd say - while I don't know if this is the state of the art now - it could definitely be a reasonable value for "today". Suppose we build 100 of these lasers - that would be 7000 MJ, and we want to figure out the largest mass. Using speed $0.1\ \mathrm{c}$, so that $\gamma \approx 1.005$ and
$$m_\mathrm{max} = \frac{2(7000\ \mathrm{MJ})}{(3.00 \times 10^8\ \mathrm{m/s})^2 \cdot 1.005 \cdot 0.1} \approx 1.54 \times 10^{-12}\ \mathrm{MJ \cdot {s^2/m^2}}$$
or $1.54 \times 10^{-12}\ \mathrm{Gg}$ (gigagrams). Taking those units down we see this is about 1.5 milligrams.
The question then becomes whether you can do anything useful with 1.5 mg of total payload, most of which will have to be taken up by the light sail - indeed, if such a light sail is possible at all. Hence whether this qualifies as "a probe" is something for which I would exercise considerable caution and mind you that I am much more of a theoretician than an engineer, so those who are more savvy with the latter may want to chime in and complete this answer. Moreover, note that this has some very optimistic hidden assumptions such as that we can reflect 100% of the laser light (impossible), and that we can keep 100% of the beam focused on the craft (this is a big issue with the real BTSS project). Hence maybe you could say that 0.15 mg might be a better target and, it doesn't then start to sound too good for the sail.
One can, of course, work the other way, too: given the energy and a craft mass, how fast can we get it? $0.1\ \mathrm{c}$ may be out, but what about if we're willing to at least send an interstellar precursor, e.g. something like the "thousand astronomical units" (TAU) that was once upon a time proposed a very long time ago. Suppose we were to take a craft mass of, say, 1 gram, or 1000 mg. Using the same equations, we can solve for $\gamma v$ by
$$\gamma v = \frac{2E}{mc}$$
so that with now $E = 7000\ \mathrm{MJ}$ and $m = 10^{-9}\ \mathrm{Gg}$, we get a $\gamma v$ of about $46\ \mathrm{km/s}$, so this is about the actual velocity. Not much better than chemical rockets, but could get you to 1000 AU - 150 000 Gm - in (noting that km/s is the same as Gm/Ms) ~3200 Ms which, while longer than a typical human lifetime of 2200 Ms (~70 years) or even a long one of 3000, is still within range of a few who'd be lucky. Still rather abysmal though esp. given what I said about this being very idealized as in the previous case.
So I'd say that, yeah, it is probably not feasible to get a space probe going with this route either. Nonetheless, I'm at least a little surprised by how and that is actually something you could perhaps at least see with your eyes that we just might, were we to deem it worthy of spending the money, loft, if not right now then quite sooner than 50 years (1577Ms). Keep in mind that "cool" can, if nothing else, be inspiration for better.
One more angle I'd point out is that in order for the lasers to be really useful, you'd ideally not want to launch this from Earth, but rather from the Moon, because of the atmosphere. Fortunately, a chemical gasdynamic laser is near-ideal for that due to the fact that it contains its own powerplant; the downside is MIRACL was a pretty big thing, and would require a lot of launch capacity to get 100 of them to the Moon. Nonetheless, it could be possible esp. with Elon Musk's BFRs - though that still is "not today".
New Horizons was the fastest man made object in space reaching 16.26 km/s after launch. After gravity assistance 23.3 km/s was reached later.
The speed of light is about 300,000 km/s. 0.001 c is 300 km/s, roughly 20 times the speed of New Horizons and 400 times the kinetic energy. Due to the rocket equation 300 km/s is impossible with present day technology.
The heaviest things we are capable of accelerating to 0.1c today are heavy atoms or small molecules.
You can always strech the definition of "probe", of course.
Is it possible to create a relativistic space probe going at least 0.1c with present day technology?
Of course! A child could do it! A child could do it!
Let's get the logarithms out of the way first. With $m_f/m_i = 20$ and ignoring special relativity we'd need an exhaust velocity $v_E$ of $0.1 \ c \ / \ \ln(20) = 0.033 \ c$.
What energy protons would we need from an ion engine for their velocity to be $0.033 \ c$?
$$E = \frac{1}{2} m v^2 = \frac{1}{2} m c^2 \left(\frac{v}{c}\right)^2$$
The mass of a proton $m_P c^2$ is about 938 MeV, so the energy would have to be
$$E = \frac{938}{2} 0.033^2 = 0.54 \text{ MeV or } 540 \text{ keV}$$
So if you built a spaceship that was 95% by mass liquid hydrogen and the other 5% were an electric powered low voltage proton RFQ linac or even just a gridded accelerator at 540 keV, you're good to go! You have space to be your vacuum pump and if you are clever the coatings of your resonators can be superconducting to minimize ohmic $I^2 R$ losses that copper would produce so you might be able to keep your power fairly low. You'll still need an ion source that makes protons and you'll have to recycle all the unionized hydrogen and protons that can't be bunched into your linac's acceptance if you use one, but them's the breaks.
If your system is having mass efficiency problems (losing hydrogen) then just crank up your RFQ to a few MeV.
See this answer to If specific impulse is directly related to exhaust velocity, would a ion post-accelerator improve the Isp of a propulsion system? for further reading.
Here's a 5 MeV RFQ (that tiny thing on the left) followed by another booster: LIGHT: A Linear Accelerator for Proton Therapy
(click for full size) left: Lawrence Berkeley Laboratory Radio Frequency Quadrupole (RFQ) right: Technician Adjusting a Radio Frequency Quadrupole (RFQ)
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