Again as the title suggests is the above true? If not what examples are there. My only background regarding these spaces have to deals with Hadamard manifolds, which is why I am seeing if its true in greater generality.
The point with Gromov hyperbolic spaces is that you have no control on their local behaviours. One way to translate this idea formally is that any bounded metric space is Gromov hyperbolic. Counterexamples to your question may be:
- Topologist's sine curve;
- An open disc minus the center.