  The generalized Proca theories with second-order equations of motion can be
healthily extended to a more general framework in which the number of
propagating degrees of freedom remains unchanged. In the presence of a
quartic-order nonminimal coupling to gravity arising in beyond-generalized
Proca theories, the speed of gravitational waves $c_t$ on the
Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological background can be equal
to that of light $c$ under a certain condition. By using this condition alone,
we show that the speed of gravitational waves in the vicinity of static and
spherically symmetric black holes is also equivalent to $c$ for the propagation
of odd-parity perturbations along both radial and angular directions. As a
by-product, the black holes arising in our beyond-generalized Proca theories
are plagued by neither ghost nor Laplacian instabilities against odd-parity
perturbations. We show the existence of both exact and numerical black hole
solutions endowed with vector hairs induced by the quartic-order coupling.
