I have heard that the punishment is that she will forever be Zonah but I have no sources
The relevant source is Mishneh Torah, Issurei Biah 18:2-3 (and from there in Shulchan Aruch, Even Haezer 6:8), that indeed such relations render her a zonah, thus making her forbidden to marry a kohen. (To be sure, as Shalom noted in comments, that's a consequence, not a punishment, and it applies even if she was raped.)
As for actual punishments: if it was in a marital relationship (and with her consent), then Rambam ibid. 12:1 states that she's liable to lashes by biblical law, although Shulchan Aruch ibid. 16:1 cites a dissenting opinion. If it was a casual relationship (and again, she consented to it), then I'm not sure: Rambam and Shulchan Aruch there say that a Jewish man who has such relations with a non-Jewish woman is subject to lashes by rabbinical law, but they don't say anything about the converse case.