This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable but not Gateaux-differentiable. By employing the limited differentiability properties of the control-to-state map, first-order necessary optimality conditions in qualified form are established, which are equivalent to the purely primal condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The paper ends with the application of the general results to a semilinear heat equation.