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While our Prophet () was still alive some sahaba have recorded hadith the most prominent example is Abdullah ibn 'Amr ibn al-'Aas, who according his own statement and a statement of abu Hurairah have been allowed (by our Prophet()) or at least has written ahadith during the life time of the Prophet (). Abduallah ibn 'Amr called his hadith collection as-Sadiqah (or as-sahifa as-Sadiqah, also read this article in Arabic about it), it is widely known because his grand son 'Amr ibn Sho'aib used to narrated and read from it. Note that it was also narrated that Anas ibn Malik and other sahaba had similar collections (this -in Arabic- thread seems to show a long list and answer your Question too). Also our Prophet () has sent a dictated paper with some rulings with Amr ibn Hazm to Yemen (See for example in sunan an-Nasa'i, a short and a long version) beside other dictate "ahadith" to people who had a bad memory.
al-Muwatta' the earliest hadith collection?
Note that Muwatta' Malik has been compiled (one could say the first release) somewhere between 148 a.H. and 163 a.H. and in worst case even later (See here). We also know that Imam abu Hanifa (died 150 a.H.) had an own hadith collection called "musnad abu hanifa مسند ابي حنيفة" which some hanafi scholars like Mulla 'Ali al-Qari have even commented, so this book must have existed at a time or what we have is falsely attributed to him. Again one could discuss whether this book has really been compiled by abu Hanifa or not. But for sure abu Hanifa had his hadith "data base" from which we find examples in the version of al-Muwtta' of his student Muhammad ibn al-Hassan a-Shaybani who after commenting of ahadith of Imam Malik, added some narrations of abu Hanifa and other scholars!