Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID? 
Is $\mathbb{Z}_{p}[G]$ a PID, where $G=(\mathbb{Z}_{p},+)$ is the additive group of the $p$-adics $\mathbb{Z}_{p}$?

I am studying a paper where the authors implicitly use that claim, but it is unclear to me. (I am a little bit embarassed by the fact that I cannot solve this myself.)

This isn't true; in fact, $\mathbb{Z}_p[G]$ is not even Noetherian.  For instance, take the augmentation ideal $I$, i.e. the ideal generated by $\{g-1:g\in G\}$.  If $I$ were finitely generated, there would be a finite subset $F\subset G$ such that $I$ is generated by the elements $g-1$ for $g\in F$.  But if $H\subseteq G$ is the subgroup generated by $F$ and $J$ is the ideal generated by the elements $g-1$ for $g\in F$, it is easy to see that the canonical quotient map $\mathbb{Z}_p[G]\to\mathbb{Z}_p[G/H]$ factors through the quotient $\mathbb{Z}_p[G]\to\mathbb{Z}_p[G]/J$.  Thus if $J$ is all of $I$, $H$ must be all of $G$.  But $G$ is not finitely generated, so this is impossible.
