Let $X$ be the set of all points in $\ell^2$ with all rational coordinates. $X$ is known to be totally disconnected, but $X$ is not zero-dimensional. For instance, the empty set does not separate the point $\langle 0,0,0,...\rangle\in X$ from the closed set $\{x\in X:\|x\|\geq 1\}$ because $\{\|x\|:x\in A\}$ is unbounded for every clopen set $A\subseteq X$.

The set $S:=\{x\in X:\|x\|=1/2\}$ separates $\langle 0,0,0,...\rangle$ and $\{x\in X:\|x\|\geq 1\}$. That is, $X\setminus S$ is the union of two disjoint open sets, one containing $\langle 0,0,0,...\rangle$, and the other containing $\{x\in X:\|x\|\geq 1\}$. Note that $S$ has no isolated points; $\overline {S\setminus \{s\}}=S$ for every $s\in S$.

My question is:

Does there exist a closed scattered separator between $\langle 0,0,0,...\rangle$ and $\{x\in X:\|x\|\geq 1\}$? A set is *scattered* if every non-empty subset has an isolated point.