Section 34: Problem 2 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
For a metric space being separable and second-countable is equivalent, at the same time if a space is second-countable and regular then it is metrizable, so, an example of a space which is completely normal, first-countable, separable and Lindelöf but not second-countable is necessary and sufficient to answer the question.
was shown to be completely normal (in fact, it is perfectly normal) and to satisfy all the countability axioms but one: it is not second-countable.