Brownian-dynamics computer simulations show a dynamical crossover in a strongly interacting colloidal suspension consisting of oppositely driven particles, wherein a uniform state transforms, with increasing driving force, into a locally demixed state characterized by strongly anisotropic stripe-like domains which are homogeneous in the direction parallel to the drive but have finite spatial extent of a double-correlation length in the transverse direction. A phenomenological dynamic density-functional theory has been proposed which accounts for such a strongly anisotropic state as arising from an instability of a homogeneous state.