Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$.  Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$.  Call a set a ball-intersection if it is an intersection of closed balls in $X$. 

Does there exist a decreasing sequence of nonempty ball-intersections in $X$ with void intersection?

If we let $S_0=\cap_i B[e_i,\pi/2]=\{x\in X: x_i\ge 0 \forall i \},$ $S_1=\{x\in S_0: x_1=0\},$ $S_2=\{x\in S_0: x_1=x_2=0\}, \cdots$, then $\cap_i S_i=\emptyset$. But  $S_i$ for $ i\neq 0$ are not ball-intersections.