In zero-dimensional Hausdorff Lindelof space each open cover has an open disjoint refinement

Question

Whether in every Zero-dimensional Hausdorff Lindelof space each open cover has a clopen pairwise disjoint refinement.

Answer 1

First refine by a clopen cover ( using zero-dimensionality) then reduce that clopen cover to a countable one. Then disjointify in the usual way.

So yes, it can be done and Hausdorff is not needed.