Driving a many-body quantum system in a periodic manner gives access to its fundamental properties, both in terms of energy spectrum and relaxation mechanisms. It also leads to important applications, as shown by Superconducting Josephson Junctions (SCJJs): Thanks to the so-called Shapiro resonances that occur in the presence of a microwave drive, SCJJs constitute metrological devices relating the drive frequency to the voltage across the junction. Here we present a detailed experimental study of an atomic analog of a driven SCJJ, based on a spinor Bose-Einstein condensate of sodium atoms. We analyze the short-time evolution of the system in terms of a slow Hamiltonian dynamics, superimposed with a rapid micro-motion. After a long-time evolution, we observe that the system may relax to a non-equilibrium steady state and exhibit a hysteretic behavior. We compare our experimental results with simple phenomenological models of dissipation, that can roughly be described as amplitude or phase damping. We find that the amplitude damping model is able to reproduce quantitatively our observations, while the phase damping model fails qualitatively in certain regimes. Our study therefore constitutes an accurate benchmark for the development of an ab initio microscopic theory of the relaxation processes in this driven many-body system.