  We extend previous results on healthy derivative self-interactions for a
Proca field to the case of a set of massive vector fields. We obtain non-gauge
invariant derivative self-interactions for the vector fields that maintain the
appropriate number of propagating degrees of freedom. In view of the potential
cosmological applications, we restrict to interactions with an internal
rotational symmetry. We provide a systematical construction order by order in
derivatives of the fields and making use of the antisymmetric Levi-Civita
tensor. We then compare with the one single vector field case and show that the
interactions can be broadly divided into two groups, namely the ones obtained
from a direct extension of the generalized Proca terms and genuine multi-Proca
interactions with no correspondence in the single Proca case. We also discuss
the curved spacetime version of the interactions to include the necessary
non-minimal couplings to gravity. Finally, we explore the cosmological
applications and show that there are three different vector fields
configurations giving rise to isotropic solutions. Two of them have already
been considered in the literature and the third one, representing a combination
of the first two, is new and offers unexplored cosmological scenarios.
