  The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic
space $\mathbb{H}^{2n+p}$ with low codimension $2\leq p\leq n-1$ under only
intrinsic assumptions remains a wide open problem. The situation is quite
different for submanifolds in the round sphere $\mathbb{S}^{2n+p}$, $2\leq
p\leq n-1$, since Florit, Hui and Zheng have shown that the codimension has to
be $p=n-1$ and then that any submanifold is just part of an extrinsic product
of two-dimensional umbilical spheres in
$\mathbb{S}^{3n-1}\subset\mathbb{R}^{3n}$. The main result of this paper is a
version for Kaehler manifolds isometrically immersed into the hyperbolic
ambient space of the result for spherical submanifolds. Besides, we generalize
several results obtained by Dajczer and Vlachos.
