Is it mentioned anywhere in all of the Mishna or the Talmud that Paul of Tarsus was a student of Rabban Gamaliel?
Acts 22:3 I am verily a man which am a Jew, born in Tarsus, a city in Cilicia, yet brought up in this city at the feet of Gamaliel, and taught according to the perfect manner of the law of the fathers, and was zealous toward God, as ye all are this day.
For more information, I would recommend reading "Paul and Gamliel", in Jacob Neusner and Bruce Chilton (eds.), In Quest of the Historical Pharisees (Baylor University Press, 2007), 175-223 - but especially p208 onwards.
Short answer: nobody named Paul (nor, for that matter, Shaul) ever gets mentioned as a student of this or any other Gamliel, although some teachings associated with Gamliel might be viewed as an influence over some Pauline passages in the NT epistles.
A few caveats: note that growing up in the dust at someone's feet does not necessarily equal being a disciple, and people may have taken more disciples than those that get named.
Secondly, the rabbinic corpus has a distinct anti-sectarian tendenz, so even if Paul really were a disciple of Gamliel's it is unlikely that he would be so named.
And thirdly, there is also a well-attested phenomenon whereby a convert or a penitent might exaggerate the nature of their former religiosity (or lack thereof). Christian scholars generally assume Paul's claim in this regard to be one such example of an exaggeration. Note that Gamliel is nowhere else mentioned by Paul - not even in his epistle to the Philippians, where he claims to have once been a Pharisee.
Although Paul of Tarsus is not mentioned by name, the Talmud in Shabbat 30b has a student of Rabbi Gamliel* who scoffs at his teachings.
Some scholars have suggested that this character might be Paul, but there is really no firm evidence to back this claim. So the answer to your question: no, Paul is not mentioned to be a student of Rabbi Gamliel in the Mishnah or Talmud. In fact, he is not mentioned at all, and Christianity seems to be a very hushed topic in the Talmud anyways.
*This Rabbi Gamliel is Gamliel I, known as "Rabban (our teacher) Gamliel."
There is a story in Shabbos 116a-b involving Raban Gamliel that according to the version of Tosafos would also involve Paul. He does not appear in the role of a student.
The Gemara reads (as corrected by old prints brought in Oz VeHadar):
אימא שלום דביתהו דרבי אליעזר אחתיה דרבן גמליאל הואי הוה ההוא פילוסופא בשבבותיה דהוה שקיל שמא דלא מקבל שוחדא בעו לאחוכי ביה אעיילא ליה שרגא דדהבא ואזול לקמיה אמרה ליה בעינא דניפלגי לי בנכסי דבי נשי אמר להו פלוגו אמר ליה כתיב לן במקום ברא ברתא לא תירות אמר ליה מן יומא דגליתון מארעכון איתנטלית אורייתא דמשה ואיתיהיבת עוון גליון וכתיב ביה ברא וברתא כחדא ירתון למחר הדר עייל ליה איהו חמרא לובא אמר להו שפילית לסיפיה דעוון גליון וכתב ביה אנא עוון גליון לא למיפחת מן אורייתא דמשה אתיתי אלא לאוספי על אורייתא דמשה אתיתי וכתיב ביה במקום ברא ברתא לא תירות אמרה ליה נהור נהוריך כשרגא אמר ליה רבן גמליאל אתא חמרא ובטש לשרגא:
The Gemara relates: Imma Shalom, the wife of Rabbi Eliezer, was Rabban Gamliel’s sister. There was a Christian philosopher [pilosofa] in their neighborhood who disseminated about himself the reputation that he does not accept bribes. They wanted to mock him and reveal his true nature. She privately gave him a golden lamp, and she and her brother came before him, approaching him as if they were seeking judgment. She said to the philosopher: I want to share in the inheritance of my father’s estate. He said to them: Divide it. Rabban Gamliel said to him: It is written in our Torah: In a situation where there is a son, the daughter does not inherit. The philosopher said to him: Since the day you were exiled from your land, the Torah of Moses was taken away and the avon gilyon was given in its place. It is written in the avon gilyon: A son and a daughter shall inherit alike. The next day Rabban Gamliel brought the philosopher a Libyan donkey. Afterward, Rabban Gamliel and his sister came before the philosopher for a judgment. He said to them: I proceeded to the end of the avon gilayon, and it is written: I, avon gilayon, did not come to subtract from the Torah of Moses, rather to add to the Torah of Moses. And it is written there: In a situation where there is a son, the daughter does not inherit. She said to him: May your light shine like a lamp, alluding to the lamp she had given him. Rabban Gamliel said to him: The donkey came and kicked the lamp, thereby revealing the entire episode.
Toasfos says:
פילוסופא... ואית דגרסי פילא סבא והוא לשון לצון ושחוק כדאמר באיכא רבתי דפלי ביהודאי פירוש ששחק ונתלוצץ
Philosopha. ... And some have the version "Pila Saba" (Paul the Elder?), and 'pila' is a language of (word meaning) scoffing and jest, like we find in Eicha rabasi "who 'pali' with the Jews", meaning that he made jest and scoffed.
While this is a beautiful source, I realized that it probably doesn't answer the question. The Raban Gamliel who was the brother-in-law of Rabbi Eliezer was Raban Gamliel of Yavneh, not Raban Gamliel the Elder. This story mentions the exile, implying that it took place after the destuction of the temple. And the Christian is quoting a written evangelical book; I'm not so familiar, but wasn't that written by Paul? For these reasons it would seem that this story is about a different Paul the Elder.
