Hadiths were collected during the lifetime of the Prophet, Sallalahu aleyh iwa sallam. I was hearing alecture by yasir qahdi, and he confirms this. This is what he said, with slight modification:

Contraty to poplular belief, the sunnah of the Prophet, sallalahu aleyi wa sallam, was written down, or at least the process was started during his sallalahu aleyi wa sallam life time.
Amr al-‘Aas, used to write down the hadeeth of the Prophet sallalahu aleyi wa sallam. Once whilst doing this, a doubt came to him as to whether he should write down everything he said, even his jokes? He went to the Prophet sallalahu aleyi wa sallam, and asked him whether he should write everything down even when he jokes? The Prophet sallalahu aleyi wa sallam pointed to his tongue and said “Uktub” write. For I swear by Allah, nothing comes from this tongue except the truth.
So Amr al-‘Aas, radiallahu used to writre down a hadeeth of the Prophet, sallalahu aleyi wa sallam during his (sallalahu aleyi wa sallam) lifetime.


Source Shiekh Yaasir Qahdis Lecture, Usool al-hadeeth, part 3.