  In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere
with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian
surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s,
$s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a
characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere
$\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss
map, and we classify spacelike surfaces and Lorentzian surfaces in the de
Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type
pseudo-spherical Gauss map. Finally, according to the causal character of the
mean curvature vector we obtain the classification of submanifolds of a
pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant
component in its spectral decomposition.
