I am looking for a (Hausdorff or better) space $X$ and a subset $A$ of $X$ that is relatively countably compact (every sequence from $A$ has an accumulation point in $X$) such that the closure of $A$ is not countably compact. It is known that in many "nice" spaces such examples do not exist (a classical case being normed spaces in their weak topology).

Edit: we have a nice Tychonoff example below, but a $T_4$ example would be nicer still.