We single out a notion of staticity for non-compact spaces which encompasses several known examples including any domain in hyperbolic space whose boundary is a non-compact totally umbilical hypersurface. For a (time-symmetric) initial data set modeled at infinity on any of these latter examples, we formulate and prove a positive mass theorem in the spin category under natural dominant energy conditions (both on the interior and along the boundary) whose rigidity statement in particular implies that no such umbilical hypersurface admits a compactly supported deformation keeping the original lower bound for the mean curvature. Joint work with S. Almaraz.