Gene drives may be used in two ways to suppress vectored diseases. Both involve engineering the drive to spread in the vector population. One has the drive directly depress vector numbers, possibly to extinction. The other leaves intact the vector population but suppresses the disease agent during its interaction with the vector. This second application may use a drive engineered to carry a genetic cargo that suppresses the disease agent. An advantage of the second application is that it is far less likely to select vector resistance to block the drive, but the disease agent may instead evolve resistance to the inhibitory cargo. However, gene drives are expected to spread so fast and attain such high coverage in the vector population that, if disease agent can evolve resistance only gradually, disease eradication may be feasible. Here we use simple models to show that spatial structure in the vector population can greatly facilitate persistence and evolution of resistance by the disease agent. We suggest simple approaches to avoid some types of spatial structure, but others may be intrinsic to the populations being challenged and difficult to overcome.