Let X be a compact, smooth manifold and Diff(X) the diffeomorphism group. The topology of Diff(X) and of the classifying space BDiff(X) are of great interest. For instance, the k-th homotopy group of BDiff(X) corresponds to smooth families over the k-sphere with fibres diffeomorphic to X. By a recent result of Bustamante, Krannich and Kupers, if X has even dimension not equal to 4 and finite fundamental group, then the homotopy groups of BDiff(X) are all finitely generated. In contrast we will show that when X is a K3 surface, the second homotopy group of BDiff(X) contains a free abelian group of countably infinite rank as a direct summand. Our families are constructed using the moduli space of Einstein metrics on K3. Their non-triviality is detected using families Seiberg--Witten invariants.