Is every zero-dimensional metrisable space ultrametrisable?

Question

Let us recall a few definitions. A topological space is

• metrisable if it is homeomorphic to a metric space.

• ultrametrisable if it is homeomorphic to an ultrametric space.

• zero-dimensional if every point has a basis of clopen neighbourhoods.

• strongly zero-dimensional if for each closed subset $F$ and each neighbourhood $U$ of $F$, there is a clopen neighbourhood of $F$ contained in $U$.

Question. Is every zero-dimensional metrisable space ultrametrisable?

If I am not mistaken, this is true for compact spaces and more generally, for Lindelöf spaces. For the general case, according to an exercise in Bourbaki [General topology, Chapter 9], a metrisable space is ultrametrisable if and only if it is strongly zero-dimensional. Moreover, it is added in a footnote that it is not known whether every zero-dimensional metrisable space is strongly zero-dimensional.

Thus the problem was apparently open when Bourbaki published his volume on general topology, but I wonder whether it has been solved since then.

Answer 1

Prabir Roy, Nonequality of Dimensions for Metric Spaces, Transactions of the American Mathematical Society Vol. 134, No. 1 (Oct., 1968), pp. 117-132, available here [PDF], has a complicated construction (which I have not gone through) of a complete metric space that is zero-dimensional but not strongly zero-dimensional.