Is a ball always connected in a connected metric space? If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?

It is easy to see the answer is no. Take the unit circle in the plane, with the metric inherited from the Euclidian metric in the plane. Then remove the "top" point (coordinates $(0,1)$). Then a "ball" (small disk) around a point close to "the top", restricted to the "circle with the top removed," will be disconnected. Here the metric space is the circle with the top point removed.
