  In this paper, we prove a Heintze-Karcher type inequality for capillary
hypersurfaces supported on various hypersurfaces in the hyperbolic space. The
equality case only occurs on capillary totally umbilical hypersurfaces. Then we
apply this result to prove the Alexandrov type theorem for embedded capillary
hypersurfaces in the hyperbolic space. In addition, we prove some other
rigidity results for capillary hypersurfaces supported on totally geodesic
plane in $\mathbb B^{n+1}_+$.
