  Let $p$ be a prime and let $G$ be a subgroup of a Sylow pro-$p$ subgroup of
the group of automorphisms of the $p$-adic tree. We prove that if $G$ is
fractal and $|G':\mathrm{st}_G(1)'|=\infty$, then the set $L(G)$ of left Engel
elements of $G$ is trivial. This result applies to fractal nonabelian groups
with torsion-free abelianization, for example the Basilica group, the
Brunner-Sidki-Vieira group, and also to the GGS-group with constant defining
vector.
  We further provide two examples showing that neither of the requirements
$|G':\mathrm{st}_G(1)'|=\infty$ and being fractal can be dropped.
