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We don't have (and more's the pity!) R' Yehudah Hanassi's autograph copy of the Mishnah (well, okay, it's a controversial issue whether he actually wrote it or only arranged it orally), or Ravina and Rav Ashi's of the Gemara, etc.
But what is the oldest known manuscript of the Mishnah, Gemara, Mechilta, or any other part of Torah Shebaal Peh? (Or perhaps there are manuscripts of Geonic writings - responsa, Halachos Gedolos, or whatever - that predate any known copies of the foundational texts?)
The oldest written Torah Sheb'al Peh (though not exactly a manuscript) would seem to be the recently-discovered mosaic of the ancient shul in Beit Shaan. The Hebrew Wikipedia article about that shul (which was destroyed 1300 years ago, and existed a few hundred before that) can be found at this link. Here is their image of the mosaic:
The text is similar to passages in Yerushalmi Sheviis. An analysis of this text can be found in Teshuvos Mishnas Yosef 1:51, and at the end of Kaftor Vaferach vol. 1 (in the 3-volume edition).
The oldest actual manuscript fragment appears to be the Cairo Genizah scroll fragment in the Cambridge University Library Genizah collection (which can be searched for ALL its wonderful things here-try "Talmud" or "ketubah" or "Rashi" for starters) studied by Professor Shamma Friedman, containing the Bavli's Chullin 101a - 105a. Opinions to the exact date vary, from "at latest 7th century(600's CE)" to Dr. Stephan Reif's estimate of around 750 CE. A picture and brief description of it is here and Professor Friedman's full JSTOR article is here. You have to sign up to read the full article, but it's free.
Please also see: An Early Fragment of "Avot de Rabbi Natan" from a Scroll מרק ברגמן, Marc Bregman, An Early Fragment of "Avot de Rabbi Natan" from a Scroll / קטע קדום של אבות דרבי נתן מתוך מגילה, Tarbiẕ / תרביץ, כרך נב, חוברת ב (טבת-אדר תשמ"ג), pp. 201-222 https://www.jstor.org/stable/23595970
Another interesting old Yerushalmi fragment from Leningrad:
A Genizah Fragment of Talmud Yerushalmi in the Antonin Collection of the Saltykov-Shchedrin Library in Leningrad on JSTOR Abraham I. Katsh, A Genizah Fragment of Talmud Yerushalmi in the Antonin Collection of the Saltykov-Shchedrin Library in Leningrad, The Jewish Quarterly Review, New Series, Vol. 71, No. 3 (Jan., 1981), pp. 181-184 https://www.jstor.org/stable/1454392
