We present a trust-region-based adaptive finite-element framework to numerically solve a class of nonsmooth PDE-constrained optimization problems. In particular, we consider the class of problems whose objective function is the sum of a smooth, possibly nonconvex, function and a nonsmooth convex function. Our method combines the robustness of inexact trust-region algorithms for nonsmooth problems with the efficiency of adaptive finite-element discretizations. Starting from a coarse mesh, the algorithm automatically refines the discretization based on reliable a posteriori error estimators for both the state and adjoint equations, systematically controlling the accuracy of the computed objective function value and gradient. This adaptive mechanism enables efficient resolution of localized phenomena and sparsity structures in the state and control variables. We demonstrate the effectiveness of our algorithm through numerical experiments on control and topology optimization examples.