We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and $\dim_{\mathbb C}X-\widetilde q(X)\leq6$ has ordinary universal cover $F\times\mathbb C^{\widetilde q(X)}$, where F is a rational homogeneous manifold and $\widetilde q(X)$ is the maximal irregularity of a connected finite étale cover.