  We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in
hyperbolic space with index of relative nullity at least one at any point. The
case when the ambient space is either the Euclidean space or the round sphere
was already studied in \cite{dksv1} and \cite{dksv2}, respectively. If the
scalar curvature is bounded from below we conclude that the submanifold has to
be either totally geodesic or a generalized cone over a complete minimal
surface lying in an equidistant submanifold of $\Hy^n$.
