  We study the local persistence probability during non-stationary time
evolutions in disordered contact processes with long-range interactions by a
combination of the strong-disorder renormalization group (SDRG) method, a
phenomenological theory of rare regions, and numerical simulations. We find
that, for interactions decaying as an inverse power of the distance, the
persistence probability tends to a non-zero limit not only in the inactive
phase but also in the critical point. Thus, unlike in the contact process with
short-range interactions, the persistence in the limit $t\to\infty$ is a
discontinuous function of the control parameter. For stretched exponentially
decaying interactions, the limiting value of the persistence is found to remain
continuous, similar to the model with short-range interactions.
