Were there any Tanaim in Bavel?

Question

In the time of the Gemara (since at least the time of Rav), there were famous Rabbis in Bavel (and, at least in the early years, there were also famous Rabbis in Israel).

But before that time, you don't hear of anyone of any importance living in Bavel. The only "Babylonian" that I know off was Hillel, and he became famous only after moving to Israel.

You may answer that while there were big Rabbis then in Bavel, they just weren't known in Israel, and since the Sanhedrin (and Rebbi - the author of the Mishna) lived in Israel, he only mentioned local Rabbis.

Yet the Yerushalmi, which was written in Israel, albeit later, mentions "Babylonians" such as Shmuel. Were there simply no big Yeshivas in Bavel in the time of the Tanaim, and if there were, why are they so rarely mentioned?

Answer 1

Yes. The last Mishna of Yebamot, as Rabbi Akiva testimonies, Nechemia Ish Bet Deli, a student of Rabban Gamliel Hazaken was in Bavel, in the city of Nehardea, there was a communication between Bavel and Erets Israel, at least in times of peace:

טז,ו [ז] אמר רבי עקיבה, כשירדתי לנהרדעא לעבר את השנה, מצאני נחמיה איש בית דלי; אמר לי, שמעתי שאין משיאין את האישה בארץ ישראל על פי עד אחד, אלא יהודה בן אוא. ונמתי לו, כן הדברים. אמר לי, אמור להם משמי, אתם יודעים שהמדינה הזאת משובשת בגייסות; מקובל אני מרבן גמליאל הזקן, שמשיאין את האישה על פי עד אחד. וכשבאתי והרצתי את הדברים לפני רבן גמליאל, שמח לדבריי, ואמר, מצינו חבר ליהודה בן אוא. מתוך הדברים נזכר רבן גמליאל, שנהרגו הרוגים בתל ארזא; והשיא רבן גמליאל הזקן את נשותיהם, על פי עד אחד. והוחזקו להיות משיאין עד מפי עד, מפי אישה ואישה מפי אישה, מפי עבד, מפי שפחה. רבי אליעזר ורבי יהושוע אומרין, אין משיאין את האישה על פי עד אחד. רבי עקיבה אומר, לא על פי אישה, ולא על פי קרובים. אמרו לו, מעשה בבני לוי שהלכו לצוער עיר התמרים, וחלה אחד מהם, והניחוהו בפונדק, ובחזירתן אמרו לפונדקית איה חברנו, ונמת להם מת וקברתיו; והשיאו את אשתו. אמרו לו, לא תהא כוהנת כפונדקית. אמר להם, ובשתהא הפונדקית נאמנת. והפונדקית הוציאה להם מקלו, ותרמילו, ומנעלו, ואפונדתו, וספר תורה שהיה בידו. ‏

Rabbi Akiva said: When I went down to Nehardea to intercalate the year, I found Nechemiah of Beit Delhi who said to me, "I heard that in the Land of Israel only Rabbi Yehudah ben Bava allows a woman to marry based on a single witness." And I said to him, "That is so". He said to me, "Tell them in my name: You know that the country is rife with soldiers; I have a tradition from Rabban Gamliel the Elder that we allow a woman to marry based on a single witness. And when I went and reported this before Rabban Gamliel he exulted in my words and said, 'We have found a colleague for Rabbi Yehudah ben Bava.' Through the conversation Rabban Gamliel recalled that people were killed at Tel Arza and Rabban Gamliel allowed their wives to marry based on a single witness, and they established a rule to allow women to marry based on a single witness, and they established a rule to allow women to marry based on [what] a witness [heard] from a witness, and based on [the testimony of] a slave, [or] based on [the testimony of] a woman, [or] based on [the testimony of] a maidservant. Rabbi Eliezer and Rabbi Yehoshua say: We do not allow a woman to marry based on [the testimony] of a single witness. Rabbi Akiva says: [We do] not [allow a woman to marry] based on [the testimony] of a woman, and not based on [the testimony] of a slave, and not based on [the testimony] of a maidservant, and not based on [the testimony] of relatives. They [the Sages] said to him: It once happened that some Levites went to Tzo'ar, the city of palm trees, and one of them fell ill on the journey, and he was brought to an inn [and left there]. And on their return they said to the woman innkeeper, "Where is our friend?" She said to them, "He died and I buried him", and they allowed his wife to marry. They said to him [Rabbi Akiva]: And shall the daughter of a priest not be as [trusted] as the woman innkeeper? He said to them: When the woman innkeeper is trusted [in this case that is because] the woman innkeeper brought out to them his staff, and his bag, and his Torah scroll that he had had with him.

Answer 2

The Mishnah in Rosh Hashanah (22b) records that they would light bonfires to tell the people of Bavel it was Rosh Chodesh. When they stopped using bonfires, they used messengers. So clearly there were people in Bavel at the time.

As for famous Tannaim, the Gemara in AZ 18b records that after R' Meir freed his sister-in-law from a brothel and the money he gave the guard to bribe the latter's superiors ran out, the guard spilled the story, at which point R' Meir was forced to flee to Bavel. Others say it was because he indirectly caused his wife to commit suicide (see Rashi there, citing Kiddushin 80b). Now, since one should only leave to a place of Torah (Avos 4:14), presumably there was an established Yeshiva that was already there.