I am looking for a space as in the title and since many very similar spaces do exist in the literature, I wonder whether someone has a reference (different from the ones I cite below) or just some remarks about this question.
A space is ccc iff any family of pairwise disjoint open sets is at most countable, and countably compact iff any countable open cover has a finite subcover. A space is separable iff it has a countable dense subset. It is immediate that a separable space is ccc.
Related facts that I know:
If one looks for a perfectly normal such space, the answer is independent of ZFC: MA + $\neg$CH implies that a perfectly normal countably compact space is compact (Weiss 1978), and this result is also compatible with CH (Eisworth 2002). Moreover, if $\diamondsuit$ holds, there is a perfectly normal, (hereditarily) separable, first countable, countably compact, non-compact space (Ostaszewski 1976).
If one relaxes first countability, such spaces do exist in ZFC, see for instance Henno Brandsma's answer to this related question: https://mathoverflow.net/questions/79021 (the given examples are Frechet-Uryson and collectionwise normal). One can also obtain separable examples (Franklin-Rajadopalan 1970).
If one wants a space which is only Tychonov but not normal, there is an example in ZFC (Bell 1989).
The related question of whether there is a countably compact, first countable, separable, non-compact regular space in ZFC is still open, as far as I know. Nyikos wrote several papers on the subject.