Consider $(\mathbf{Z}\times[0,1])/(\mathbf{Z}\times\{0\})$, infinitely many spokes attached at a single point. Given a finite basis, the only points uniquely characterized by distances from it are the points on the same spokes as the basis. So this is infinite-dimensional.

Consider $(\mathbf{Z}\times[0,1])/(\mathbf{Z}\times\{0\})$, infinitely many spokes attached at a single point. Given a finite basis, the only points uniquely characterized by distances from it are the points on the same spokes as the basis. So this space is infinite-dimensional.