  The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for
the study of hyperbolic geometry in one and in several complex variables. In
the present paper, we turn our attention to the quaternionic unit ball B. We
prove a version of the Schwarz-Pick lemma for self-maps of B that are slice
regular, according to the definition of Gentili and Struppa. The lemma has
interesting applications in the fixed-point case, and it generalizes to the
case of vanishing higher order derivatives.
