  We consider the vectorial analogue of the thin free boundary problem
introduced in \cite{CRS} as a realization of a nonlocal version of the
classical Bernoulli problem. We study optimal regularity, nondegeneracy, and
density properties of local minimizers. Via a blow-up analysis based on a Weiss
type monotonicity formula, we show that the free boundary is the union of a
"regular" and a "singular" part. Finally we use a viscosity approach to prove
$C^{1,\alpha}$ regularity of the regular part of the free boundary.
