The mechanism of peristaltic transport of an incompressible viscoelastic fluid by means of an infinite train of sinusoidal waves travelling along the wall of the duct is studied in the case of a plane flow. The main assumptions are that the relevant Reynolds number is small enough to neglect inertia forces, and that the ratio of the wavelength and the channel height is large, which implies that the pressure is constant over the cross-section. For sufficiently small values of the ratio of the wave amplitude and the mean height of the channel, details of the fluid motion are studied analytically within a second-order approximation with respect to the amplitude ratio. Under these conditions the integral constitutive equation of finite linear viscoelasticity is relevant. Particular attention is given to the pressure–discharge characteristics of the peristaltic pump and to the pumping efficiency. The results are influenced by specific values of the complex viscosity of the fluid, which can be determined using standard rheometers. In general, the rate of discharge turns out to be a non-monotonic function of the wave speed. This leads to an optimal wave speed, for which the memory of the fluid particles extends over several wave periods. From an energetic point of view, relatively small wave speeds are the best, where the fluid changes its state slowly such that the memory and with it the elasticity of the fluid do not influence the flow field at all. As the dimensionless memory parameter tends to zero, the analytical results reduce to the well-known case of a Newtonian fluid.