Note that any metric with unique infinite geodesics on $\mathbb R^2$ has this property. In particular, hyperbolic plane as noted by Heather Macbeth. It also includes Minkowski plane with smooth ball and all complete negatively curved Riemannian metrics on $\mathbb R^2$.
So it is better to ask:
Is it true that the function $p$ is the only possible? ($p(a,b,c)=0,1,$ or $2$ when the triangle inequality for a,b,c, correspondingly, fails, turns into equality, or is strict.)
Is it true that space has to be homeomorphic to $\mathbb R^2$?
Edit: You condition implies existence of both-side infinite minimizing geodesic through any pair of points. It is likely that such geodesic must be unique but I do not see it immediately. If true, applying cone construction you get a map of 3-simplex in your space, if it is nondegenerate (i.e. the image does not coinsides with image of its boundary) then it would be easy to get a triple of small numbers $a,b,c$ such that $p(a,b,c)=\infty$ thus we arrive to a contradiction. Thus the space must be 2-dimensional and with unique infinite geodesics --- I bet it should be homeomorphic to $\mathbb R^2$...