  For each prime $p$ and each positive integer $d$, we construct the first
examples of second countable, topologically simple, $p$-adic Lie groups of
dimension $d$ whose Lie algebras are abelian. This answers several questions of
Gl\"ockner and Caprace-Monod. The proof relies on a generalization of small
cancellation methods that applies to central extensions of acylindrically
hyperbolic groups.
