There are old ZFC examples due to Eric van Douwen that satisfy all the properties in the title, except for being of cardinality $2^{\aleph_0}$, so the answer to the title question is YES if the Continuum Hypothesis (CH) is assumed, but I am asking for an example of which does not use any axiom beyond the ZFC axioms.

It would be especially nice if it were first countable, like Eric's examples, and normal, like one of his examples. A Souslin tree with the interval topology qualifies. I have been able to weaken CH to "stick" [which says that there is a family of $\aleph_1$ countable subsets of an uncountable set, such that every uncountable subset contains a member of the family] and I also have an example if $\mathfrak b = \aleph_1$ but no ZFC example.