Is someone has a non-Jewish father but a Jewish mother, this obviously makes them a Jew. But does it make them a mamzer?
The halacha is that he is not a mamzer.
See Yebamot 16b:
On account of the slaves of Solomon,' the reason is quite intelligible, because he may hold the opinion that the child of a heathen or a slave who had intercourse with a daughter in Israel is a bastard.
Said Rabina: From this it follows that the 'son of your daughter' who derives from a heathen is called 'thy son'. Does this imply that Rabina is of the opinion that if a heathen or a slave had intercourse with a daughter of Israel the child i considered fit! -Though he is admittedly no bastard neither is he considered fit; he is rather regarded as a tainted Israelite.
And the law is that if an idolater or a slave had cohabited with the daughter of an Israelite the child [bor from such a union] is legitimate, both in the case of a married, and in that of an unmarried woman.
SA EH 4, 19:
עובד כוכבים ועבד.... ואם באו על בת ישראל, בין פנויה בין אשת איש, הולד כשר, ופגום לכהונה:
NJ or slave... if they had relationship with a Jewish woman, celibate or married, the child is Kasher, but not for marriage with a Kohen (the daughter)*
- see this post for the last rule and for a little more extended explanation.