We prove existence and uniqueness of the mild solution of an infinite dimensional, operator valued, backward stochastic Riccati equation. We exploit the regularizing properties of the semigroup generated by the unbounded operator involved in the equation. Then the results will be applied to characterize the value function and optimal feedback law for an infinite dimensional, linear quadratic control problem with stochastic coefficients. Moreover we shall show that it covers the second variation equation arising in the optimal control of the stochastic heat equation in an interval (see [M. Fuhrman, Y. Hu, and G. Tessitore, C. R. Acad. Sci. Paris Ser. 1 Math., 350 (2012), pp. 683-688] and [M. Fuhrman, Y. Hu, and G. Tessitore, Appl. Math. Optim., 68 (2013), pp. 181-217]).