Rabbi Jacob Bar Abba asked Abaye: Is it permitted to a disciple in a district under his Master's jurisdiction to give a ruling that was as authoritative as those contained in the Scroll of Fast-Days, which is a written and generally accepted document? - Thus, the other replied, said Rav Joseph: Even on the question of the permissibility of eating an egg with kutha, which I have been asking him throughout the lifetime of Rav Huna, R`Hisda gave me no decision. (Eruvin 62b)
אלו ימים אסורין להתענות להכי נקט מגילת תענית שלא היתה דבר הלכה כתובה בימיהן אפילו אות אחת חוץ ממגילת תענית ולהכי קרי לה מגלה: (Rashi)
... There was no Halacha written at that time, even one letter, except for Megilat Taanit...
The Iggeret exists in two recensions, a "French" and a "Spanish" recension. The "French" recension is in Aramaic, while the "Spanish" recension contains a higher proportion of Hebrew. The two recensions appear to differ on the question of whether the Mishnah was recorded in writing by Rabbi Judah haNasi. The Spanish recension definitely says that it was. The French recension appears to say that it was not, and this was the traditional view among Ashkenazi Jews.
See R. Binyamin Menashe Levin's introduction to his edition of the Iggeret Rav Sherira where he discusses the two recessions, and notes that they correspond to a general dispute between the Franco-German scholars and the Spanish scholars about whether Rabbi Yehuda HaNassi wrote down the Mishna. The Spanish scholars include Mevo HaTalmud, and Rambam (introduction to Mishneh Torah). The French scholars include Rashi, Semag (introduction), and the Sefer HaTerumah (245), quotes his mentor R. Isaac of Dampierre that:
בימי התנאים והאמוראי' שהיו חכמי מדע ונבוני לב ולא היו שוכחין דבר לפיכך לא היו כותבין לא תלמוד ולא תפלות משום דברים שבעל פה אי אתה רשאי לכותבן
That is, in the time of the Tannaim and Ammoram they did not write down any works of the Oral Law, such as the Talmud, due to the prohibition to write down the Oral Law.
Similarly, R. Isaac Stein (15th century) writes explicitly in his commentary to Semag, that the Mishna (and Talmud) were not originally written, but merely compiled (and relayed verbally):
שמעתי ממקצת לומדים לשם שאמרו איך שרבי כתב המשניות מששה סדרים ורב אשי כתב הגמרא. ואמרתי להם שאינו כן רק סדרום ולא כתבום
For several examples where the Spanish version clearly indicates that he wrote the Mishna, while the French version ommits the indicative phrases, see here.
For example, the Spanish version reads:
ולא הוה חד מן הראשונים דכתב מדעם עד סוף יומי דרבינו הקדוש
While the French version reads:
ועד סוף יומייהו דרבנן ליכא דכתב מדעם כל עיקר
Rashash (Sh'vuot 4a) notes as well that according to Rashi (s.v. UMishna lo zaza), R. Yehuda HaNassi did not write down the Mishna. Similarly, the Hikrei Lev (Vol. I OH 12) notes that according to Rashi in several places it is clear that the Mishnayot were not written until well after the time of the compilation of the Talmud.
Prof. Sussman notes there that Rashi implies this in many other places as well.
As R. Dr. Levin notes, later scholars continued this dispute. R. Yaakov Hagiz (introduction to Ets HaHayyim), R. Yosef Hazan (Hikrei Lev there), and Shadal argued that the Mishna was not written down, while R. Yisrael Moshe Hazan (R. Yosef's grandson), and later R. Zev Yaavets argued that the Mishna was written down by R. Yehuda HaNassi.