How Old Are Perakim?

Two weeks ago, in a class on the practice of blowing shofar on Rosh Chodesh, there was an issue of the appropriateness of the first Mishna in the context of its perek - it looked like it belonged to the previous perek. One fellow student suggested that the division of mishnayot into perakim was late - perhaps as late as the Geonim. Dr. Steinfeld asked what basis he had to make such a statement, but proceeded even assuming that the division into perakim was that late.

My thoughts immediately turned to several yerushalmis which refer to the first perek, etc.. Anyway, while the pesikta (the part of the Mishna quoted before the two dots) is Geonic, the division into perakim is known to the Amoraim. Further, they even referred to them as we do - by the first words of the first Mishna of the perek. This was known as far back as Rav - a first generation Amora with quasi-Tanna status.

Here is a good prooftext - yerushalmi gittin 47a. Perek 8 halacha 5:

רב הונא בשם רב כל הדא פירקא דרבי מאיר
חוץ משינה שמו ושמה שם עירו ושם עירה.
רבי מנא בעי קומי רבי יסא אף המגרש?
אמר ליה לית פירקא דידכון אלא דידן

Rav Huna cited Rav: {The opinions expressed in} this entire perek is in accordance with Rabbi Meir
Except for {the law of when the sofer} changed his name, her name, his city's name, or her city's name {where everyone agrees}.
Rabbi Mana asked before Rabbi Yassa: Also {Perek} HaMegaresh? {Besides this Perek, HaZorek?}
He said to him: Not your perek but only ours {HaZorek}.