  Let $X$ be a compact K\"ahler fourfold with klt singularities and vanishing
first Chern class, smooth in codimension two. We show that $X$ admits a
Beauville-Bogomolov decomposition: a finite quasi-\'etale cover of $X$ splits
as a product of a complex torus and singular Calabi-Yau and irreducible
holomorphic symplectic varieties. We also prove that $X$ has small projective
deformations and the fundamental group of $X$ is projective. To obtain these
results, we propose and study a new version of the Lipman-Zariski conjecture.
