  Given $H\in [0,1)$ and given a $C^0$ exterior domain $\Omega$ in a
$H-$hypersphere of $\mathbb{H}^3,$ the existence of hyperbolic Killing graphs
of CMC $H$ defined in $\overline{\Omega}$ with boundary $ \partial \Omega $
included in the $H-$hypersphere is obtained.
