  Despite the many applications of rate-independent systems, their regularity
theory is still largely unexplored. Usually, only weak solution with
potentially very low regularity are considered, which requires non-smooth
techniques. In this work, however, we directly prove the existence of
H\"older-regular strong solutions for a class of rate-independent systems. We
also establish further assertions about higher regularity of our solutions. The
proof proceeds via a time-discrete Rothe approximation, careful elliptic
regularity estimates in the discrete situation and evolutionary techniques.
