  We develop a notion of rank one properly convex domains (or Hilbert
geometries) in the real projective space. This is in the spirit of rank one
non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank
one isometries for Hilbert geometries and characterize them as being equivalent
to contracting elements (in the sense of geometric group theory). We prove that
if a discrete subgroup of automorphisms of a Hilbert geometry contains a rank
one isometry, then the subgroup is either virtually cyclic or acylindrically
hyperbolic. This leads to several applications like infinite-dimensionality of
the space of quasi-morphisms, counting results for conjugacy classes and
genericity results for rank one isometries.
