Non-perfectly-normal Tychonoff spaces meeting Theorem 2.5 hypotheses

Identify whether there exists a Tychonoff space X that is not perfectly normal and nevertheless contains a countably infinite subset N such that (i) N is not nowhere dense in X, (ii) X \ N is dense in X, and (iii) every singleton subset of N is a zero set in X, implying T"(X) is not closed under uniform limits.

Background

Theorem 2.5 shows that if X contains a countably infinite subset N with specific properties, then T"(X) is not closed under uniform limit and hence is not isomorphic to any C(Y).

The authors provide several perfectly normal examples (e.g., metric spaces, normed linear spaces, Sorgenfrey line) and ask whether a similar example exists outside the class of perfectly normal spaces.