Is there a non-metrizable topological space for which any countably compact subset is compact?

Any Lindelöf non-metrizable Hausdorff space will do (EDIT: you need that the space is C-closed as well, see below), but more generally, a space is called isocompact iff every closed countably compact subset of X is compact, cl-isocompact iff the closure of a countably compact subset is compact, and C-closed iff any countably compact subset is closed. This paper by Cho and Park contains various results on (cl)-isocompactess, as well a other references:

http://www.mathnet.or.kr/mathnet/kms_tex/978637.pdf

And this one by Ismail and Nyikos has good informations on C-closed spaces:

http://www.sciencedirect.com/science/article/pii/0166864180900279

In addition to the above examples, it might be of interest that functional analysis is awash with such spaces. For example, the space of distributions on a compact interval or that of functions analytic on a closed domain in the complex plane, both with their natural topologies. More generally, any Silva space has this property. A further ubiquitous source of examples is the dual of a separable Banach space with the bounded weak $\ast$ topology. A suitable reference would be Köthe's classic on topological linear spaces.

I think any compact, first countable, non-metrizable Hausdorff space suffices, since then any countably compact set is closed and hence compact. In Steen and Seebach's Counterexamples in Topology, #95, the weak parallel line topology, has these properties.