Denoting with H^n the n-dimensional hyperbolic space, we show that constant mean curvature hypersurfaces in H∧n×R with small boundary contained in a horizontal slice P are topological disks, provided they are contained in one of the two half-spaces determined by P. This is the analogous in H∧n×R of a result in R^3 by A. Ros and H. Rosenberg. The proof is based on geometric and analytic methods: from one side the constant mean curvature equation is a quasilinear elliptic PDE on manifolds, to the other the specific geometry of the ambient space produces some peculiar phenomena. This talk is based on a joint work with Barbara Nelli.