  In this paper, we study the functional convergence in law of the fluctuations
of the derivative martingale of branching random walk on the real line. Our
main result strengthens the results of Buraczewski et. al. [Ann. Probab., 2021]
and is the branching random walk counterpart of the main result of Maillard and
Pain [Ann. Probab., 2019] for branching Brownian motion.
