  We investigate fundamental properties of the proximal point algorithm for
Lipschitz convex functions on (proper, geodesic) Gromov hyperbolic spaces. We
show that the proximal point algorithm from an arbitrary initial point can find
a point close to a minimizer of the function. Moreover, we establish
contraction estimates (akin to trees) for the proximal (resolvent) operator.
Our results can be applied to small perturbations of trees.
