This question was motivated by this MO-question asking about the example of a hereditarily disconnected metrizable separable space, which is not the union of countably many totally disconnected subspaces.

Let us recall that a topological space $X$ is

$\bullet$ hereditarily disconnected if every connnected subspace of $X$ has cardinality $\le 1$;

$\bullet$ totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$.

Now let us describe a family of "canonical" hereditarily disconnected subspaces of the Hilbert cube.

Let $\mathbb I=[0,1]$ and $\mathbb I^{<\omega}=\bigcup_{n\in\omega}\mathbb I^n$.

Given a family $D=(D_x)_{x\in\mathbb I^{<\omega}}$ of dense subsets of $\mathbb I$, consider the subspace $$H_D=\mathbb I^\omega\setminus\bigcup_{n\in\omega}\{x\in\mathbb I^\omega:x(n)\in D_{x{\restriction}_n}\}=\bigcap_{n\in\omega}\{x\in\mathbb I^\omega:x(n)\notin D_{x{\restriction}_n}\}.$$ It is easy to see that the space $H_D$ is hereditarily disconnected.

If $D_x=\mathbb I\cap\mathbb Q$ for all $x\in\mathbb I^{<\omega}$, then $H_D=(\mathbb I\setminus\mathbb Q)^\omega$ is zero-dimensional.

Problem. Is there a family $D=(D_x)_{x\in\mathbb I^{<\omega}}$ of dense subsets of $\mathbb I$ such that the hereditarily disconnected space $H_D$ is not the countable union of totally disconnected subspaces?

Example. Choose two disjoint countable dense sets $Q_0,Q_1$ in $\mathbb I$ and take any family $(D_x)_{x\in\mathbb I^{<\omega}}$ of countable dense subsets of $\mathbb I$ such that $D_\emptyset=Q_0$, $D_{(x)}=Q_0$ for all $x\in \mathbb I\setminus Q_1$, $D_{(x)}=Q_1$ for all $x\in Q_1$, and $0\notin D_x$ for all $x\in\bigcup_{n\ge 2}\mathbb I^n$. Then the space $H_D$ is not totally disconnected. More precisely, its subspace $\{x\in H_D:\forall n\ge 2\;(x(n)=0)\}$ is not totally disconnected (which can be shown by analogy with Kuratowski-Knaster fan).