  A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly
hereditarily self-similar of index $p$ if every non-trivial finitely generated
closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree.
We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of
dimension less than $p$ that are strongly hereditarily self-similar of index
$p$. Moreover, we show that a solvable torsion-free $p$-adic analytic pro-$p$
group of dimension less than $p$ is strongly hereditarily self-similar of index
$p$ if and only if it is isomorphic to the maximal pro-$p$ Galois group of some
field that contains a primitive $p$-th root of unity. As a key step for the
proof of the above results, we classify the 3-dimensional solvable torsion-free
$p$-adic analytic pro-$p$ groups that admit a faithful self-similar action on a
$p$-ary tree, completing the classification of the 3-dimensional torsion-free
$p$-adic analytic pro-$p$ groups that admit such actions.
