  We study actions of higher rank lattices $\Gamma<G$ on hyperbolic spaces, and
we show that all such actions satisfying mild properties come from the rank-one
factors of $G$. In particular, all non-elementary actions on an unbounded
hyperbolic space are of this type. Our results also apply to lattices in
products of trees, so that for example Burger--Mozes groups have exactly two
non-elementary actions on a hyperbolic space, up to a natural equivalence.
