This paper develops second-order necessary conditions for nonsmooth infinite-dimensional optimization problems with Banach space-valued equality and real-valued inequality constraints. Another constraint in the form of a closed convex set is also present. The objective function is the maximum over a parameter of functions f(t, z) that are Lipschitz in z and upper semicontinuous in t. The inequality constraints g(s, z) depend on a parameter s. The technique we use is a generalization of that of Dubovitskii and Milyutin. The second-order conditions obtained here are in terms of a certain function sigma that disappears when the parameters and a certain set that derives from the given convex set are absent. The presence of the function sigma and that set is due to the envelope-like effect discovered by Kawasaki.