This work is concerned with nonlinear parameter identification in partial differential equations subjectto impulsive noise. To cope with the non-Gaussian nature of the noise, we consider a model withL1 fitting. However, the nonsmoothness of the problem makes its efficient numerical solution challenging.By approximating this problem using a family of smoothed functionals, a semismooth Newtonmethod becomes applicable. In particular, its superlinear convergence is proved under a second-ordercondition. The convergence of the solution to the approximating problem as the smoothing parametergoes to zero is shown. A strategy for adaptively selecting the regularization parameter basedon a balancing principle is suggested. The efficiency of the method is illustrated on several benchmarkinverse problems of recovering coefficients in elliptic differential equations, for which one- andtwo-dimensional numerical examples are presented.