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?

There are even complete path-connected metric spaces that contain a point $x$ such that no ball around $x$ is connected, for example
$$ \{\langle x,0\rangle \mid x\ge 1\} \cup
\{\langle 1,y\rangle \mid 0\le y\le 1 \} \cup
\{\langle x,\tfrac1x \rangle \mid x\ge 1 \} \cup
\bigcup_{n=3}^\infty \{ \langle x,\tfrac1n \rangle \mid 2 \le x \le n \}
 $$
No ball of finite radius around $\langle 2,0\rangle$ is connected.

And here's a sketch of a complete and connected (but not path-connected) subspace of $\mathbb R^2$ where no ball whatsoever, around any point, is connected:

Each of the gaps in the Cantor set on the left will eventually be contracted -- but the smaller the gap, the farther to the right will this happen.
