We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph $\mathit{G}=(\mathit{V},\mathit{E})$: an individual is attached to each site $\mathit{x}\in \mathit{V}$, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate $\mathit{\lambda }>0$; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution μ such that $\mathit{\mu }(\mathit{t},\infty )={\mathit{t}}^{-\mathit{\alpha }}\mathit{L}(\mathit{t})$, where $1/2<\mathit{\alpha }<1$ and $\mathit{L}(·)$ is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if $|\mathit{V}|<2+(2\mathit{\alpha }-1)/[(1-\mathit{\alpha })(2-\mathit{\alpha })]$, then the infection does not survive for any λ; and if $|\mathit{V}|>1/(1-\mathit{\alpha })$, then, for every λ, the infection has positive probability to survive.