We consider a spatial model of a susceptible-infected-susceptible (SIS) process in which individuals are organized into demes of identical population size. The demes themselves are structured on a spatial lattice. An individual can be in one of three states: susceptible, infected with a drug-resistant strain, or infected with a wild-type strain. Infections spread both within demes and between adjacent demes. Drug treatment enters at the level of demes. Some demes are treated permanently with drug, such that only the resistant strain can infect the individuals inside these demes. The treated demes can be spatially correlated, representing larger structures such as hospitals where drug treatment is employed, or treatment can be spatially uncorrelated. When treated demes are spatially correlated, both the resistant strain and the wild type coexist for timescales many orders of magnitude larger than when drug treatment is not spatially correlated. We analyze and simulate the process in many different parameter regimes and characterize both the initial transients and a quasi-steady state in which both infections exist in similar frequencies and in similar spatial locations for long time periods. We postulate that the magnitude of the extinction time for the drug-resistant strain, which is overall less fit than the wild type, can be approximated by the extinction time for an Ornstein-Uhlenbeck process. These spatial effects may explain the long-term coexistence between wild type infections and drug-resistant strains that has been observed in the United States over the last 70 years.