We introduce a general class of causal dynamic discrete-time nonlinearities which have certain monotonicity and Lipschitz continuity properties. In particular, the discretizations of a large class of continuous-time hysteresis operators obtained by applying the standard sampling and hold operations belong to this class. It is shown that closing the loop around a power-stable, linear, infinite-dimensional, discrete-time, single-input, single-output system, subject to an input nonlinearity from the class under consideration and compensated by a discrete-time integral controller, guarantees asymptotic tracking of constant reference signals, provided that ( a) the positive integrator gain is sufficiently small and (b) the reference value is feasible in a very natural sense. We apply this result in the development of sampled-data low-gain integral control for exponentially stable, regular, linear, infinite-dimensional, continuous-time systems subject to input hysteresis.