E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
Headlong; in reckless confusion. From the players of pallmall, who rush heedlessly to strike the ball. The pall is the ball (Italian, palla), and the mall is the mallet or bat (Italian, maglia; Latin, malleus). Sometimes the game is called pall mall; and sometimes the ground set apart for the game, as Pall Mall, Londo.
It is not quite certain that pell-mell is the same compound word as pall-mall.