In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space H-s(m-1)(-1) subset of E-s+1(m) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in H-s(n+1) (-1) subset of E-s+1(n+2) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For n = 2, we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in H-2(4) (-1) subset of H-2(m-1) (-1) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface M-t(n) of the pseudo-hyperbolic space H-t(n+1) (-1) subset of E-t+1(n+2) has biharmonic pseudo-hyperbolic Gauss map.