  Let $p$ be a prime. We say that a pro-$p$ group is self-similar of index
$p^k$ if it admits a faithful self-similar action on a $p^k$-ary regular rooted
tree such that the action is transitive on the first level. The self-similarity
index of a self-similar pro-$p$ group $G$ is defined to be the least power of
$p$, say $p^k$, such that $G$ is self-similar of index $p^k$. We show that for
every prime $p\geqslant 3$ and all integers $d$ there exist infinitely many
pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite
uniform pro-$p$ groups of self-similarity index greater than $d$. This implies
that, in general, for self-similar $p$-adic analytic pro-$p$ groups one cannot
bound the self-similarity index by a function that depends only on the
dimension of the group.
