Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a geodesic is a globally length-minimizing path, whereas a local geodesic is a locally length-minimizing path.
I suspect the answer is, in general, negative (so that there may exist local geodesics which are not geodesics) but I haven't been able to come up with an example to see this. Also, if a counterexample exists, can it be taken to be a Riemannian manifold?