Consider numbers of the form $10^k + 1$. We can look at the prime factorisation of these numbers and note that the smallest such number that has a repeated prime factor is $10^{11} + 1 = 11^2\cdot{}23\cdot{}4093\cdot{}8779$. We can use this to form a number of the form $a = a_1a_2\dots{}a_ka_1a_2\dots{}a_k$ such that $a$ is a perfect square (the smallest such number is $1322314049613223140496 = 36363636364^2$).
The next smallest number of the form $10^{k} + 1$ with a squared prime factor is $10^{21} + 1 = 7^2\cdot{}11\cdot{}13\cdot{}127\cdot{}2689\cdot{}459691\cdot{}909091$ which can be used to form perfect squares with repeated digits in a similar fashion. Following that comes $10^{33} + 1$, again divisible by $121$ and then $10^{39} + 1$, divisible by $169$.
OEIS contains the sequence A086981, for which $s(n)$ comprises the smallest value of $k$ for which $10^k + 1$ is divisible by $p_n^2$, where $p_n$ is the $n$th prime number. Obviously numbers of our special form are not divisible by $2, 3$, or $5$ so the sequence starts $0, 0, 0, 21, 11, 39$.
For some primes $> 5$, the sequence contains a zero, indicating (quoting) no such $k$ exists. The first entry that is not trivially zero is the $11^{\mathrm{th}}$, corresponding to 31. Following that are 37, 41 and 43. The next entry is $1081$ and indeed $10^{1081} + 1$ is divisible by 2209.
Does the "no such $k$" wording mean that necessarily no such $k$ exists, or that no known $k$ exists? I have checked for e.g. divisibility by $41^2$ up to $k$ = several million just for fun.