  This paper is devoted to the study of the differentiability of solutions to
real-valued backward stochastic differential equations (BSDEs for short) with
quadratic generators driven by a cylindrical Wiener process. The main novelty
of this problem consists in the fact that the gradient equation of a quadratic
BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO
martingales. We show some applications to the nonlinear Kolmogorov equations.
