Among the lesser known songs recorded by the Beatles in the spring or summer of 1960, they did a "jazz standard" called "I Will Always Be in Love with You". According to one source I found, it was first done by somebody named Morton Downey in the late '20s, but it's been speculated that the Beatles learned it from the version by Fats Domino.
This song is an example of The Beatles performing songs that predate the rock 'n' roll era, which are mostly pop tunes on the jazzy side, and reflect the influence of the music their parents knew. In particular, there's some possibility that Paul's father, who performed in a jazz band, taught him the song- or it could very well have been John's mother, who knew some chords on the banjo.
From the sounds of it, John was trying to imitate Elvis.
Unfortunately, for the life of me, I cannot find any recorded version available to listen to that predates the Beatles version anywhere, so I guess if you wanna hear it, you'll have to buy some Fats Domino collection... which makes me dubious about it being called a "standard". I managed to hear at least a sample of it on Allmusic's entry for Bear Family's Out of New Orleans, which I recommend for completists like me. I'm not so sure if the Beatles were trying to copy it, what with the Elvis stutter, but I don't see why they couldn't have learned it from his version...
Of course, if you're desperate, there are a couple covers on YouTube made after 1960.