We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph $H$, there exists $ε> 0$ such that every $n$-vertex graph with no pivot-minor isomorphic to $H$ contains two sets $A, B$ of vertices such that $|A|, |B| \ge εn$ and $A$ is complete or anticomplete to $B$.