I apologize for the stupid question, but I am getting a bit crazy about this.
Consider a Banach space $X$ and a sequence of nested closed balls $(B_n)_n$, i.e. $B_{n+1} \subset B_n$. Let $r_n$ be the radius of $B_n$, for every $n \in \mathbb N$. Question: does the limit $$ \lim_n r_n $$ always exist?
In principle, I would say: "Yes, of course: the sequence $r_n$ must be monotonically decreasing, hence the limit exists. In particular, if it is $0$ then the intersection $\bigcap_ n B_n$ is a singleton while, if $r_n \to r$, then $\bigcap_ n B_n$ is a closed ball of radius $r$."
I am now worried about monotonicity: consider, for instance, these examples. Something related also here and here (note that in the last link the existence of the limit is the hypothesis). Also the post of t.b. here (in particular the first lines) may be relevant.
Thanks.
Except in the trivial case of the zero space, being Banach prevents pathological examples such as you link to.
More precisely, in a Banach space with at least one nonzero vector, we cannot have $B(x,R)\subseteq B(y,r)$ with $R>r$.
You can see this by considering the two balls restricted to a line that contains $x$ and $y$, with the induced metric. (In the case $x=y$ you need to assume that the space is not the zero space in order to choose such a line). The line is isometric to $\mathbb R$, and the balls intersect it at balls of the same radii on the line. And certainly on the real line, a larger ball cannot be contained in a smaller one.