This is credited to Don Raye and John Jacob Niles. It was first recorded by Peggy Lee in 1959 as the B-side of her single Hallelujah, I Love Him So.
Cliff recorded it on December 11, 1961, and it was released on May 5, 1962 as his second single without his backing group the Shadows. It reached No 2 in Britain.
I am not sure if this performance is live, but it is definitely a different recording.