Although Amalek was a descendant of Eisav (Esau) , Eisav had other descendants who were simply classified as Edomites and not Amalekites.
As opposed to Amalek which we are supposed to blot out the rememberance of Amalek under the sky (end of Dvarim/Deut. chapter 25) it says of the other descendants of Eisav, namely the Edomites
Thou shalt not abhor an Edomite; for he is thy brother.... The children that begotten of them shall enter into the congregation of Hashem in the third generation. (Dvarim/Deut. 23 verse 8 and 9).
With this in mind, see the Talmud Baba Batra page 21 where the laws of Amalek on one occasion were supposed to have been applied to the Edomites. A mistake occurred because of the negilence of the Rabbi of Yoav, so the full force of the law was not applied.
See there for details.
The story there in the Talmud fits much better with the view that there is a non-Biological Amalek (although some people have tried to claim that sometimes Edomite means all descendants of Eisav while in the context of the Baba Batra page 21 it means specifically the Biological Descendants of Amalek).