We prove that the volume of a free boundary minimal surface $\Sigma^k \subset B^n$ where is a geodesic ball in Hyperbolic space $H^n$ is bounded from below by the volume of a geodesic k-ball with the same radius as $B^n$. More generally, we prove analogous results for the case where the ambient space is conformally Euclidean, spherically symmetric, and the conformal factor is nondecreasing in the radial variable. These results follow work of Brendle and Fraser-Schoen, who proved analogous results for surfaces in the unit ball in $R^n$. This is joint work with Brian Freidin.