We investigate a non-Markovian analogue of the Harris contact process in Z(d): an individual is attached to each site x is an element of Z(d), and it can be infected or healthy; the infection propagates to healthy neighbours just as in the usual contact process, according to independent exponential times with a fixed rate lambda; nevertheless, the possible recovery times for an individual are given by the points of a renewal process with heavy tail; the renewal processes are assumed to be independent for different sites. We show that the resulting processes have a critical value equal to zero. (C) 2018 Elsevier B.V. All rights reserved.