  On the one hand, we construct a continuous family of non-isometric proper
CAT(-1) spaces on which the isometry group ${\rm Isom}(\mathbf{H}^{n})$ of the
real hyperbolic $n$-space acts minimally and cocompactly. This provides the
first examples of non-standard CAT(0) model spaces for simple Lie groups.
  On the other hand, we classify all continuous non-elementary actions of ${\rm
Isom}(\mathbf{H}^{n})$ on the infinite-dimensional real hyperbolic space. It
turns out that they are in correspondence with the exotic model spaces that we
construct.
