Existence of a CFL $L$ such that $\sqrt{L}$ is not CFL
There is an example, and $L = \{a^nb^na^{2m}b^ka^k \mid n,m,k \in \mathbb{N}\}$ does the trick. We get that $\sqrt{L} = \{a^nb^na^n \mid n \in \mathbb{N}\}$, which is a standard example of a non-context-free language.
To elaborate a bit on how to get there: CFLs can express that two numbers are the same, but not that three numbers are the same. So I want the square-root operation to introduce another equality, as it seems predisposed to do so.