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.