Oh for sure. We already had complex social relationships that involved lying when we were only homo erectus and likely incapable of speech and were hunting full grown elephants and hippopotami(yeah, simple stone tools against those monsters required some serious teamwork). I think that creating a social face for those you DON’T know, though, had to come about once we were in a situation where there were people we interacted with that we didn’t know. Hunter/gatherer societies generally still operated with too much of a cohesion for you to truly be “fake”.