  We consider a minimal extension of the Standard Model which advocates a dark
neutrino sector charged under a hidden $U(1)^\prime$. We show that neutrino
masses can arise radiatively in this model. The observed values are compatible
with a light dark sector below the electroweak scale and would imply new heavy
fermions which may be testable in the next generation of beam dump searches at
DUNE, NA62 and SHIP.
