This is what This Day in Music said about Martin:
[Martin] was a member of the legendary Nashville A-Team, playing guitar on hits ranging from Roy Orbison’s ‘Oh, Pretty Woman’, Marty Robbins’ ‘El Paso’ and Loretta Lynn’s ‘Coal Miner’s Daughter’. During a 50-year career, Martin backed such names as Elvis Presley, Buddy Holly, Woody Guthrie, Arlo Guthrie, Johnny Cash, Patsy Cline, Joan Baez and J. J. Cale.
He solos at the :43 mark of Red Foley’s Crawfish Hole. Notes elsewhere on the Internet say that Martin used a custom double-neck Bigsby guitar, which probably is the one is playing in this clip.