Disclaimer: my knowledge of $p$-adic analysis/geometry is minimal.
Consider a smooth, complete curve $C$ of genus $g$ over $\mathbb{C}_{p}$, denote by $J$ its Jacobian and consider the embedding $C\hookrightarrow J$ with respect to some base point. Let $T$ be the tangent space of $J$.
While the exponential map is defined only locally, the logarithm $\operatorname{log}:J\to T$ is defined everywhere, and the kernel are the torsion points $J^{\rm tor}$ (this is stated on the first page of Fontaine's Preseque $C_p$-représentations without proof or references, if you know better references I'd appreciate it). Furthermore, the composition $C\to J\to T=\mathbb{C}_{p}^{g}$ can be constructed directly by applying Coleman's integration to the holomorphic forms on $C$ (this seems to be folklore in the literature: again, if you know a precise reference I'd appreciate it).
Question 1: is the image of $C\to J\to T=\mathbb{C}_{p}^{g}$ closed for the $p$-adic topology?
This would be obvious if $\mathbb{C}_{p}$ was locally compact, but this is not the case. For the same reason, the question has a positive answer if we replace $\mathbb{C}_{p}$ with a finite extension of $\mathbb{Q}_{p}$.
The answer is obviously yes if $g\le 1$, so the question is really for $g\ge 2$. My impression is that $C\to T$ cannot be too badly behaved (e.g. by Manin-Mumford the fibers are finite), but I can't put my finger on it.
Let us now address a related problem (easier? more difficult? who knows). There is a splitting (at least topological) $J=J[p']\times J^{(p)}$ where $J[p']$ is the prime-to-$p$ torsion and $J^{(p)}$ is the $p$-divisible subgroup of $J$ (again, this is stated in Fontaine's paper). We have then a factorization
$$ J\to J^{(p)} \to T $$ of the logarithm, where we kill only $J[p']$ (as opposed to $J^{\rm tor}$).
The projection $J\to J^{(p)}$ is fairly easy to write down locally. If $n$ is prime with $p$, the $p$-adic expansion of $1/n$ defines an homomorphism $m_{1/n}:J^{(p)}\to J^{(p)}$ which is an inverse of multiplication by $n$. For every point $j\in J$, there exists an $n$ prime with $p$ and a neighbourhood $U\subset J$ such that $nU\subset J^{(p)}$; in this neighbourhood, the projection is just the composition
$$ m_{1/n}\circ n:U\to J^{(p)} \to J^{(p)} $$
By increasing $n$, we get larger open subsets where $m_{1/n}\circ n$ is defined, and their union covers $J$.
Question 2: is the image of $C\to J \to J^{(p)}$ closed for the $p$-adic topology?
Let us rephrase the questions in more concrete terms. Choose a point of $T$ outside the image, and let $o\in J$ be a point mapping to it. Using $o$ as origin, the fact the point is outside of the image correspond to the fact that $C\cap J^{\rm tor}=\emptyset$. Question 1 then asks if there exists an $\varepsilon>0$ such that $C$ does not intersect an $\varepsilon$-fattening of $J^{\rm tor}$. This is reminiscent of Bogomolov's conjecture. Question 2 is analogous, but we look at $\varepsilon$-fattenings of $J[p']$ rather than $J^{\rm tor}$.
Edit. I have found a result which is relevant, so I'm adding it to the question. Raynaud, in the paper where he proves the Manin-Mumford conjecture Courbes sur une variété abélienne et points de torsion, also proves the following.
Theorem (Raynaud) Let $R$ be a complete DVR of mixed characteristic $(0,p)$ with residue field $k$. Assume that $R$ is unramified over $p$, i.e. $p\in R$ is a uniformizing parameter, and that $k$ is algebraically closed, e.g. the completion of the maximal unramified extension of $\mathbb{Z}_{p}$.
Let $\mathcal{X}$ be a smooth relative curve over $R$ of genus $\ge 2$, and $\mathcal{X}\subset\mathcal{J}$ the relative jacobian. For every $a\in\mathcal{J}(R)$, the number of points of $(\mathcal{X}+a)(k)$ which lift to $(\mathcal{X}+a)(R)\cap p\mathcal{J}(R)$ is finite, with a bound not depending on $a$.
