  We study the maximal Salem degree of automorphisms of K3 surfaces via
elliptic fibrations. By generalizing \cite{EOY14}, we establish a
characterization of such maximum in terms of elliptic fibrations with infinite
automorphism groups. As an application, we show that any supersingular K3
surface in odd characteristic has an automorphism the entropy of which is the
natural logarithm of a Salem number of degree $22$.
