gn.general topology - Countable paracompactness, normality and locally countable open covers

(repost from the topology Q&A board)

I have a (T_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x in X has a neighbourhood which intersects only countably many members of the cover) has a locally finite refinement.

My suspicion is that the answer is a resounding no, but every time I try to construct a counterexample it starts to seem more plausible.

If the answer does turn out to be yes I'd love to know if it generalises from aleph_0 to arbitrary cardinals.