Let $B$ be a real Banach Space whose dimension is at least $2$, and let $S$ be a subset of $B$ that is an open ball. Is the complement of $S$ (with respect to $B$) always connected?
Idea
One could then perhaps make use of the facts that every $2$-dimensional Banach space is homeomorphic to the Euclidean plane and that - for the Euclidean plane - my question has an obvious "YES" answer (since every open ball is the interior of a circle).
Two-dimensional case
Let $C$ be a circle enclosing the ball $S$. For each point $p\in S$, consider the line through $p$ and the center of $S$. A segment of this line connects $p$ to $C$. Since $C$ itself is path-connected, $B\setminus S$ is path-connected.
General case
Given $p,q\notin S$, apply the above to a two-dimensional affine space containing $p,q$, and the center of $S$.