Simple $p$-adic Lie groups with abelian Lie algebras

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 Glöckner and Caprace-Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.