Could be, although not that particular completion. The second chorus was rare and I'm kind of unsure about "shot a man". Can't edit previous comment but should have just put it as:
"Dashing through the snow - - on a pair of broken skis - - -"
"Down the hills we go - - - Crashing into trees!"
"The snow is turning red - - I think I'm almost dead - -"
"And now I'm in the hospital with stitches in my head! Oh, -"
"Jingle bells, Batman smells, Robin laid an egg! - - -"
"Batmobile lost a wheel and the Joker played ballet! HEY!"
"Jingle bells, Batman smells, Granny got a gun, - - -"
"? ? ?, and shot a man in 1931! HEY!!!"