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

A decomposition theorem for singular K\"ahler spaces with trivial first Chern class of dimension at most four