Regular $T_2$ space which is not completely regular.

This answer has a complete description of such a space, due to John Thomas and published in A Regular Space, Not Completely Regular, The American Mathematical Monthly, Vol. 76, No. 2 (Feb., 1969), pp. 181-182.

These are counterexamples, which may be hepful for you:

Deleted Tychonoff Corkscrew

Hewitt’s Condensed Corkscrew

Thomas’s Corkscrew

Tychonoff Corkscrew