Wijsman hyperspaces of non-separable metric spaces

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Given a metric space X,ρ, consider its hyperspace of closed sets CL(X) with the Wijsman topology τW₍ᵨ₎. It is known that CL(X),τW₍ᵨ₎ is metrizable if and only if X is separable and it is an open question by Di Maio and Meccariello whether this is equivalent to CL(X),τW₍ᵨ₎ being normal. In this paper we prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then CL(X),τW₍ᵨ₎ is not normal. We also solve some questions by Cao, Junnilla and Moors regarding isolated points in Wijsman hyperspaces.