  We study perturbations of massive and massless vector fields on a
Schwarzschild black-hole background, including a non-minimal coupling between
the vector field and the curvature. The coupling is given by the Horndeski
vector-tensor operator, which we show to be unique, also when the field is
massive, provided that the vector has a vanishing background value.
  We determine the quasi-normal mode spectrum of the vector field, focusing on
the fundamental mode of monopolar and dipolar perturbations of both even and
odd parity, as a function of the mass of the field and the coupling constant
controlling the non-minimal interaction. In the massless case, we also provide
results for the first two overtones, showing in particular that the
isospectrality between even and odd modes is broken by the non-minimal
gravitational coupling.
  We also consider solutions to the mode equations corresponding to quasi-bound
states and static configurations. Our results for quasi-bound states provide
strong evidence for the stability of the spectrum, indicating the impossibility
of a vectorization mechanism within our set-up. For static solutions, we
analytically and numerically derive results for the electromagnetic
susceptibilities (the spin-1 analogs of the tidal Love numbers), which we show
to be non-zero in the presence of the non-minimal coupling.
