Uniquely geodesic and CAT(0) spaces? Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.

Question : Is a finite dimensional metric space, uniquely geodesic if and only if it is CAT(0) ?

In the case of a negative answer :
- Is CAT(0) assumption necessary ?  Is it sufficient ?
- What are the classical counter-examples ?
- Is there a slight additive assumption for having a positive answer ?
 A: The CAT(0) asumption is sufficient: it implies that any two points are connected by a unique geodesic segment. This is well-known and follows from the definitions.
However, as pointed out by HenrikRüping, the CAT(0) asumption is not necessary, you can for instance perturb the hyperbolic plane by putting a small lump of positive curvature near a point and the resulting metric will not be CAT(0) but will still be uniquely geodesic.
You might however get a positive result in this direction if you add some smoothness and a (fairly strong) topological asumption, namely, that your space is the universal cover of a torus. Then you can use the result by Burago and Ivanov here that any Riemannian metric without conjugate point on a torus is flat. If a metric is uniquely geodesic then it is very close to having no conjugate points, and by their result it has to be flat, hence CAT(0). So as far as I can see it's conceivable that a uniquely geodesic distance on the universal cover of an $n$-dimensional torus is flat, hence CAT(0) (?)
A: Here is another counterexample. Let us endow the Euclidean space $V=\mathbb{R}^n$ with the distance induced by the usual $\ell^p$-norm. Then, if $1<p< \infty$, then $V$ is uniquely geodesic. However, $V$ is CAT(0) if and only if $p=2$. (The proofs of these statements are very easy; as far as I remember, they may be found in the book by Bridson and Haefliger on metric spaces of non-positive curvature).  
A: A good counterexample is the Teichmuller space of a closed oriented surface $S$. It is uniquely geodesic by Teichmuller's theorem, but it is not $CAT(0)$.
