We introduce 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 retrieves a recent result by Souam to the effect that no such umbilical hypersurface admits a compactly supported deformation keeping the original lower bound for the mean curvature. A key ingredient in our approach is the consideration of a new boundary condition on spinors which somehow interpolates between chirality and MIT bag boundary conditions. Joint work with S. Almaraz (arXiv:2206.09768, to appear in ASNS Pisa).