Generalization is a main issue for current audio deepfake detectors, which struggle to provide reliable results on out-of-distribution data. Given the speed at which more and more accurate synthesis methods are developed, it is very important to design techniques that work well also on data they were not trained for. In this paper we study the potential of large-scale pre-trained models for audio deepfake detection, with special focus on generalization ability. To this end, the detection problem is reformulated in a speaker verification framework and fake audios are exposed by the mismatch between the voice sample under test and the voice of the claimed identity. With this paradigm, no fake speech sample is necessary in training, cutting off any link with the generation method at the root, and ensuring full generalization ability. Features are extracted by general-purpose large pre-trained models, with no need for training or fine-tuning on specific fake detection or speaker verification datasets. At detection time only a limited set of voice fragments of the identity under test is required. Experiments on several datasets widespread in the community show that detectors based on pre-trained models achieve excellent performance and show strong generalization ability, rivaling supervised methods on in-distribution data and largely overcoming them on out-of-distribution data.