The paper considers a two-person zero-sum discrete gunner-evader game GAMMA, which takes place on a semi-infinite line. The game is modeled as a recursive game with an infinite number of states. The theory of such games is far from being complete, and it is not even known whether they always have a solution. Even when they do have a solution, the optimal or epsilon-optimal strategies for the players may require a knowledge of past actions. It is shown that GAMMA has a solution and that the players have stationary optimal or epsilon-optimal strategies.