  We derive the profile of a vector field coupled to matter on a static and
spherically symmetric background in the context of generalized Proca theories.
The cubic Galileon self-interaction leads to the suppression of a longitudinal
vector component due to the operation of the Vainshtein mechanism. For quartic
and sixth-order derivative interactions, the solutions consistent with those in
the continuous limit of small derivative couplings correspond to the branch
with the vanishing longitudinal mode. We compute the corrections to
gravitational potentials outside a compact body induced by the vector field in
the presence of cubic, quartic, and sixth-order derivative couplings, and show
that the models can be consistent with local gravity constraints under mild
bounds on the temporal vector component. The quintic Galileon interaction does
not allow regular solutions of the longitudinal mode for a rapidly decreasing
matter density outside the body.
