We construct a space having the properties in the title, and with the same technique, a countably compact $T_2$ topological group which is not absolutely countably compact.
@article{118744, author = {Jerry E. Vaughan}, title = {A countably compact, separable space which is not absolutely countably compact}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {36}, year = {1995}, pages = {197-201}, zbl = {0833.54012}, mrnumber = {1334426}, language = {en}, url = {http://dml.mathdoc.fr/item/118744} }
Vaughan, Jerry E. A countably compact, separable space which is not absolutely countably compact. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) pp. 197-201. http://gdmltest.u-ga.fr/item/118744/
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