I see that it is forbidden for a Jewish girl to get married to a non-Jew (E"H 16.1) but why is she forbidden to have relations with him without marriage?
(I guess the source might be that she becomes a zona)
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Indeed a Jewish non-married woman is forbidden to have relations with non-Jews. The key sources are Vayikra 19:29
אַל־תְּחַלֵּ֥ל אֶֽת־בִּתְּךָ֖ לְהַזְנוֹתָ֑הּ וְלֹא־תִזְנֶ֣ה הָאָ֔רֶץ וּמָלְאָ֥ה הָאָ֖רֶץ זִמָּֽה׃
Do not degrade your daughter and make her a harlot (zona, see below), lest the land fall into harlotry and the land be filled with depravity.
and Devarim 23:18
לֹא־תִהְיֶ֥ה קְדֵשָׁ֖ה מִבְּנ֣וֹת יִשְׂרָאֵ֑ל וְלֹֽא־יִהְיֶ֥ה קָדֵ֖שׁ מִבְּנֵ֥י יִשְׂרָאֵל׃
No Israelite woman shall be a prostitute (kedesha, see below), nor shall any Israelite man be a prostitute.
R Azriel Schreiber explains based on these verses
The Torah itself most certainly forbids any form of prostitution and/or sexual promiscuity among Jews, whether or not both partners involved are Jewish. The classic commentary of Rashi (on the first verse above) explains the word “Zonah” to mean an unmarried woman who is available to many sexual partners. This is whether or not she receives payment for this act.
In the second verse, we find that this prohibition applies equally to both men and women. Rashi here explains that a Kedeshah is a woman who is “Hefker” – not given boundaries, and is set aside to be a Zonah, as we explained above. [...] From these verses, we see that a Zonah is any woman who has multiple sexual partners outside of the context of marriage, and that the state of being a Zonah is a direct prohibition from the Torah.
Thus, if either party is not committed to a monogamous relationship, then it would be the Torah prohibition of Zenut. (Please note, that one also needs to take into consideration the laws of Nidda – there is the Torah prohibition of Niddah.)
See also From where do we know that sexual intercourse with a non-jewish woman is forbidden? on MY - first answer aims to show the interdiction is rabbinic (although it refers to a Jew with a non-Jewish woman). See also here and there.
Devarim 7, 3:
ולא תתחתן בם בתך לא תתן לבנו.
1. You shall not intermarry with them,
2. Don't give your daughter to a gentile man.
The Gemora Avoda zara 36b says clearly: אלא דאורייתא עובד כוכבים הבא על בת ישראל דמשכה בתריה אבל ישראל הבא על העובדת כוכבים לא ואתו אינהו גזור אפי' ישראל הבא על העובדת כוכבים -
A gentile that has a one-off relationship (without formal marriage) with a Jewess is forbidden from the Torah since through this he gets her to follow his ways.
Chinuch Mitzva 27 in his list of Mitzva written in the Torah
שלא נתחתן עם האמות, וכתב הרמב"ם זכרונו לברכה (פ"יב מהל' איסורי ביאה ה"א) לא עם שבעה עממים ולא עם שאר האמות, שנאמר (דברים ז ג) ולא תתחתן בם. וענין החתנות הוא, שיתן בתו לבנו או בנו לבתו, וכמו שבאר הכתוב (שם) בתך לא תתן לבנו ובתו לא תקח לבנך. וכל שכן המזדוג עמהם, שהוא בעצמו בכלל האסור,
This mitzva is not to marry the nations which the Rambam explains even the nations aside from the 7 Canaanite nations. Chasnus means giving his child over as a partner to the gentile's child and how much more so one who chooses him/herself to have relations with them where he/she personally is included in the prohibition.
Kiddushin (marriage) does not work with a gentile so there is no marriage whatsoever it just looks like marriage so he has no connection to her and she does not require a Divorce. This is evident in Kiddushin 66a.
By relations with a gentile she is called a זונה only in that she is not allowed to marrying a Kohen but she can still marry a Yisroel. This clear from the Rambam here as it says in Yevamos 45a.
The prohibition of ויקרא יט, כט) אל תחלל את בתך להזנותה) of a regular Jewess, is for the father to give his daughter to an Old man like Rabbi Eliezer says in Sanhedrin 76a, whom we follow Lehalacha מיי' פ"כא מהל' איסורי ביאה הלכה כו , סמ"ג עשין מח , טור ושו"ע אה"ע סי' ב סעיף ט.
And the prohibition of לֹא־תִהְיֶ֥ה קְדֵשָׁ֖ה מִבְּנ֣וֹת יִשְׂרָאֵ֑ל וְלֹֽא־יִהְיֶ֥ה קָדֵ֖שׁ מִבְּנֵ֥י יִשְׂרָאֵל׃ is Like Onkles which says that A woman should not marry a Canaanite slave (who's accepted mitzvos as he's had Tevila and mila see Yevamos 46a) and a man should likewise not marry a Canaanite slavewoman (who's accepted mitzvos as she's had Tevila see Yevamos 46a). This is the reason why a Mamzer can marry a slave woman as he is already a product of an illegitimate relationship in Kiddushin 59a, see Rashi there, and Rambam Issurei bia 15,3. and Shulchan Aruch Even Haezer 4,20.
So that is why these 2 Pesukim are clearly not Halachically referring to a relationship with a Gentile contrary to what some Rabbis mistakenly preach.