Is there a Christian equivalent of the Muslim hadith, but for narrations relating to Jesus? Muslim here; I'm not particularly familiar with Christianity (so bear with me please).  In Islam, we have hadith:


  A hadith is one of various reports describing the words, actions, or habits of the Islamic prophet Muhammad.


Question: Is there a Christian equivalent of the Muslim hadith, but for narrations relating to Jesus?

I'm aware of narrations relating to Jesus in the Bible, such as curing leprosy:


  And Jesus put forth His hand and touched him, saying, “I will: be thou clean.” And immediately his leprosy was cleansed. -- Matthew 8:3


But this isn't exactly equivalent, as a hadith is ordinarily of the form "Muhammad said/did this", and is separate from the Qur'an.

Searching for hadith on this site reveals Does Christianity have any other books besides the Bible that the disciples use for following?, but this is about books, rather than narrations.  Searching for "christian equivalent of hadith" or "christian narrations of jesus" wasn't helpful.
The Gospels might be considered akin to the hadith in how they are used. The Catholic and Orthodox sects of Christianity read the Gospels in a similar way to the hadith, as the start of a divinely inspired oral and written tradition which involved building a centuries-long consensus over interpretation and practice. The Protestant sects read the Gospels in a very different way. Each person reads the Gospels alone (sola scriptura) and forms his own judgments about them, sometimes with the assistance of a church community.

There are also a few orthodox "hadith" which are not in the Gospels but are found in early Christian writings and other documents. These are called "agrapha". However, agrapha are treated as akin to the more dubious hadith. They were first assembled in a single book by the German Alfred Resch in 1889. Since then they have been followed by some other textual discoveries, but none of these really contribute to the tradition of the Church in any of its mainstream forms.