We construct a continuous family of non-isometric minimal proper CAT(-1) spaces on which the isometry group Isom(Hⁿ) of the hyperbolic n-space acts minimally and cocompactly by isometries. This provides the first examples of non-standard CAT(0) model spaces for simple Lie groups. We also classify all continuous non-elementary actions of Isom(Hⁿ) on the infinite-dimensional real hyperbolic space.