Abstract Whether Wada basins of strange nonchaotic attractors (SNAs) can exist has been an open problem. Here we verify the existence of Wada basin for SNAs in a quasiperiodically forced Duffing map. We show that the SNAsʼ basins are full Wada for a set of parameters of positive measure. We identify two types of SNAsʼ Wada basins by the basin cell method. It suggests that SNAs cannot be predicted reliably for the specific initial conditions.