  In this work we present a new family of properly embedded free boundary
minimal hypersurfaces of revolution with circular boundaries in the horizon of
the $n$-dimensional Schwarzschild space, $n\geq3$. In particular, we answer a
question proposed by O. Chodosh and D. Ketover \cite{CK} on the existence of
non-totally geodesic minimal surfaces in the 3-dimensional Schwarzschild space.
