Generalized linear models are popular for modelling a large variety of data. We consider variable selection through penalized methods by focusing on resistance issues in the presence of outlying data and other deviations from assumptions.We highlight the weaknesses of widely-used penalized M-estimators, propose a robust penalized quasilikelihood estimator, and show that it enjoys oracle properties in high dimensions and is stable in a neighbourhood of the model. We illustrate its finite-sample performance on simulated and real data.