  We show that large classes of non-arithmetic hyperbolic $n$-manifolds,
including the hybrids introduced by Gromov and Piatetski-Shapiro and many of
their generalizations, have only finitely many finite-volume immersed totally
geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic
submanifolds of dimension at least $2$ that are maximal, i.e., not properly
contained in a proper geodesic submanifold of the ambient $n$-manifold. The
proof is a mix of structure theory for arithmetic groups, dynamics, and
geometry in negative curvature.
