Exponent of self-similar finite $p$-groups

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 \mid 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.