Let X be a complex K3 surface, Diff (X) the group of diffeomorphisms of X and Diff 0 (X) the identity component. We prove that the fundamental group of Diff 0 (X) contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg–Witten invariants. The moduli space of Einstein metrics on X is used as a key ingredient in the proof.