The paper deals with expansivity of a nonsmooth dynamical system, in which all trajectories that remain within a certain threshold of each other must be identical. Explicit knowledge of the rates of separation is useful for numerical calculations and shadowing arguments. Known are sufficient conditions of expansivity for finite dimensional dynamical systems [\textit{P. Diamond, P. E. Kloeden, V. Kozyakin} and \textit{A. Pokrovskij}, Bull. Aust. Math. Soc. 51, No. 2, 301-308 (1995; Zbl 0826.58028)]. In this paper sufficient conditions are presented for the exponential expansivity of a discrete-time dynamical system that is generated by a continuous mapping of a Banach space into itself. This mapping need not be differentiable or invertible and the subset under consideration need not be invariant. The main result is to show that these conditions are satisfied by shift of nonlinear functional differential equations formed by Lipschitz perturbations of a linear functional differential equation with a hyperbolic equilibrium point.