  Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with
$\mathrm{dim}(G) < p$, we show that the self-similar actions of $G$ on regular
rooted trees can be studied through the virtual endomorphisms of the associated
$\mathbb{Z}_p$-Lie lattice. We explicitly classify 3-dimensional unsolvable
$\mathbb{Z}_p$-Lie lattices for $p$ odd, and study their virtual endomorphisms.
Together with Lazard's correspondence, this allows us to classify 3-dimensional
unsolvable torsion-free $p$-adic analytic pro-$p$ groups for $p\geqslant 5$,
and to determine which of them admit a faithful self-similar action on a
$p$-ary tree. In particular, we show that no open subgroup of
$SL_1^1(\Delta_p)$ admits such an action. On the other hand, we prove that all
the open subgroups of $SL_2^{\triangle}(\mathbb{Z}_p)$ admit faithful
self-similar actions on regular rooted trees.
