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.