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 Poincaré 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 Poincaré 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.