We have performed magnetization measurements on a Pb thin film perforated by a square array of antidots in a broad temperature range. For temperatures below 6 K, we observe magnetization jumps at low fields. In the film with the antidot lattice, the region in the field-temperature plane, where the jumps are observed, is substantially enlarged but the relative jump size is smaller compared to a reference film without any nanostructuring. Also, the width of the magnetization loops for the perforated film is much smaller than that for the plain film at very low temperatures. These results can be understood by taking into account the existence of a multiterrace critical state induced by the antidot array. The size distributions of the magnetization jumps do not follow the power-law decay expected in the case of self-organized criticality.