Let $X$ be a compact Kähler manifold admitting a metric with negative holomorphic sectional curvature. It was proved by Wu-Yau (when $X$ is projective) and Tosatti-Yang (in general) that the canonical bundle $K_X$ is ample. In particular, any smooth submanifold $Y \subset X$ has $K_Y$ ample, too. I will explain that any possibly singular subvariety $Y$ of $X$ is of general type, in the sense that any smooth model of $Y$ has big canonical bundle. This enables to check the validity of Lang's conjecture about complex hyperbolic manifolds in this particular setting.