An on- and off-lattice analog of the driven lattice gas has been studied by molecular-dynamics simulation. In the model, particles move on a two-dimensional lattice potential under a constant driving force. They also interact with each other by an attractive square-well pair potential. A heat bath removes as heat the work done by the driving field. The case of zero field recovers equilibrium two-dimensional lattice and continuous-space results. With nonzero field and a strong lattice potential, the system is comparable to the driven lattice gas. As in the driven lattice gas, the anisotropic single-strip configuration persists to higher kinetic energy as the field strength increases. In the case of zero lattice strength, the model reduces to an off-lattice two-dimensional driven square-well fluid. Single-strip steady states are not observed off lattice.