  We consider the Cauchy problem for the (non-linear)
Maxwell-Charged-Scalar-Field equations with spherically symmetric initial data,
on a sub-extremal Reissner--Nordstr\"{o}m or Schwarzschild exterior space-time.
We prove that the solutions are bounded and decay at an inverse polynomial rate
towards time-like infinity and along the black hole event horizon, provided the
charge of the Maxwell equation is sufficiently small. This condition is in
particular satisfied for small data in energy space that enjoy a sufficient
decay towards the asymptotically flat end. Some of the decay estimates we prove
are arbitrarily close to the conjectured optimal rate in the limit where the
charge tends to zero, according the heuristics present in the physics
literature. Our result can also be interpreted as a first step towards the
stability of Reissner--Nordstr\"{o}m black holes for the gravity coupled
Einstein--Maxwell-Charged-Scalar-Field model. This problem is closely connected
to the understanding of strong cosmic censorship and charged gravitational
collapse in this setting.
