This article presents a distributed model predictive control algorithm for a scenario of spacecraft formation flying, namely, spacecraft platooning in a leader–follower formation. This spacecraft formation application can be regarded as a cooperative system composed of several spacecraft with a common goal, for which a relative translational model is considered connecting the different agents. A distributed algorithm is proposed that solves a 1-hop optimization problem in order to estimate the optimal control action sequence, including also relevant constraints to protect and deal with the overall system. The algorithm is then analyzed in terms of uniqueness and convergence of the optimal control action. The proposed methods are validated with realistic simulation results, showing that all vehicles demonstrate reliable performance following a given trajectory or goal in a formation, while satisfying all the considered constraints.