  We study an interplay between delay and discontinuous hysteresis in dynamical
systems. After having established existence and uniqueness of solutions, we
focus on the analysis of stability of periodic solutions. The main object we
study is a Poincar\'e map that is infinite-dimensional due to delay and
non-differentiable due to hysteresis. We propose a general functional framework
based on the fractional order Sobolev--Slobodeckij spaces and explicitly obtain
a formal linearization of the Poincar\'e map in these spaces. Furthermore, we
prove that the spectrum of this formal linearization determines the stability
of the periodic solution and then reduce the spectral analysis to an equivalent
finite-dimensional problem.
