Is $\pi$ periodic in any numeral system?

$\pi$ is irrational - that was settled hundreds of years ago. That implies that the expression of $\pi$ to any integer base $b$ will be aperiodic. If you have some other kind of numeral system in mind, please edit your question accordingly.

No. In order for $\pi$ to be periodic in base $k$, it must be true that $\pi \equiv m(\pi) \pmod{k}$ for some integer $m$.

By definition of mod, this means that $m(\pi) = \pi + nk$ $\Rightarrow$ $\pi = nk/(m-1)$, which is rational. Since we know that $\pi$ is irrational, we get a contradiction.

In fact you can apply the same argument for all irrational numbers. You can conclude that any irrational number is non-periodic in $k$.