Despite your admonition to not mention oral traditions that are not mentioned in any book of the Bible, I have to say that the whole Bible is made up of oral traditions that were eventually written down! During the first century of the Church oral tradition is all they had, other than the Old Testament.
As for an oral tradition NOT mentioned in Scripture, I would offer the belief that the New Testament is Sacred Scripture and the Word of God! Certainly, nowhere will you find Jesus ever mentioning anything in Scripture about writing anything down. The only time we see Him associated with writing is when he wrote on the ground when the woman caught in adultery was standing before Him.
Nowhere will you find any of the authors of the New Testament suggesting that they themselves were in fact writing Sacred Scripture! It was the Church (Catholic) that first acknowledged this. This is an oral teaching, I believe, that is held dear by all Christians. Sola Scriptura, is an oral teaching that is held only by Protestants.
Fr. Vincent Serpa, O.P.