Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : Every finitely-presented group is geodesic, i.e. acts properly, cocompactly and isometrically on a geodesic space (for example its Cayley graph related to a finite generating set).
Definition : A group is uniquely geodesic if it acts properly, cocompactly and isometrically on a uniquely geodesic space.
Question : Are torsion-free finitely presented groups, uniquely geodesic ? (see the next remark)
Remark : we add the assumption "with a proper action on a contractible CW complex so that the action is cocompact on each skeleton" (see Misha's comment). In addition, if groups with $\infty$-dimensional classifying space (as Thompson groups), give obvious counter-examples, we add the assumption "with a finite-dimensional classifying space". Perhaps the "cocompact" assumption is not relevant in the $\infty$-dimensional case, and should be replaced by something else.
Remark : The Baumslag-Solitar groups $BS(m,n)$ are torsion-free, finitely generated (with $2$-dim. classifying space) but not CAT(0) for $m \ne n$. Perhaps they are counter-examples, I don't know...
This is an optional part about continuously uniquely geodesic groups :
Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints.
Remark : A continuously uniquely geodesic space is contractible.
Remark : A proper uniquely geodesic space is continuously uniquely geodesic, but a complete uniquely geodesic space is not necessarily continuously uniquely geodesic: see here page 38 (3.13 and 3.14)
Definition : A group is continuously uniquely geodesic if it acts properly, cocompactly and isometrically on a continuously uniquely geodesic space.
Question : Is a uniquely geodesic (finitely presented) group also continuously uniquely geodesic ?
Remark: In my opinion, the continuously uniquely geodesic (finitely presented) groups are very convenient with operator algebras conjectures as the Baum-Connes conjecture.