In this vague form your question has an easy answer: take two copies of the hyperbolic plane and identify them along a half-space. 

In fact there are more "natural" (and more complicated) objects which answer your question in the negative, called hyperbolic buildings. As in the second example they are contractible polygonal complexes locally modeled the hyperbolic plane but the branching pattern is along the 1-skeleton of a regular tesselation (for example that by right-angled pentagons). Some of these spaces also admit discrete cocompact actions. You can look at this paper of Frédéric Haglund and Frédéric Paulin and its references for more information : https://arxiv.org/abs/math/9812167.