In this paper a bundle method for nonconvex nonsmooth optimization in infinite-dimensional Hilbert spaces is developed and analyzed. The algorithm requires only inexact function value and subgradient information. Global convergence to approximately stationary points is proved, where the final accuracy depends on the error level in the function and subgradient data. The method is then applied to an optimal control problem governed by the obstacle problem. For adaptively controlling the inexactness, implementable conditions are developed, first on a general level and then for the concrete case of a FEM discretization for optimal control of an obstacle problem. Numerical results are presented.