While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a feature
that has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a L´evy-driven
spatio-temporal Ornstein-Uhlenbeck process by randomly varying its rate parameter to model both short-range and longrange
dependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable for
real applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoretical
properties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is also
used to illustrate how the memory of the process is related to the probability distribution of the rate parameter. We
develop a simulation algorithm for the compound Poisson case which can be used to approximate other L´evy bases. The
generalised method of moments is used for inference and simulation experiments are conducted with a view towards
asymptotic properties.