  Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a
faithful, self-similar action on the $p$-ary rooted tree. We prove that if the
set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then
$G$ is a finite $p$-group with exponent at most $p^n$. This applies in
particular to power abelian $p$-groups.
