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Physical-layer security is regaining traction in the research community, due to the performance boost introduced by deep learning classification algorithms. This is particularly true for sender authentication in wireless communications via radio fingerprinting. However, previous research efforts mainly focused on terrestrial wireless devices while, to the best of our knowledge, none of the previous work took into consideration satellite transmitters. The satellite scenario is generally challenging because, among others, satellite radio transducers feature non-standard electronics (usually aged and specifically designed for harsh conditions). Moreover, the fingerprinting task is specifically difficult for Low-Earth Orbit (LEO) satellites (like the ones we focus in this paper) since they orbit at about 800Km from the Earth, at a speed of around 25,000Km/h, thus making the receiver experiencing a down-link with unique attenuation and fading characteristics. In this paper, we propose PAST-AI, a methodology tailored to authenticate LEO satellites through fingerprinting of their IQ samples, using advanced AI solutions. Our methodology is tested on real data -- more than 100M I/Q samples -- collected from an extensive measurements campaign on the IRIDIUM LEO satellites constellation, lasting 589 hours. Results are striking: we prove that Convolutional Neural Networks (CNN) and autoencoders (if properly calibrated) can be successfully adopted to authenticate the satellite transducers, with an accuracy spanning between 0.8 and 1, depending on prior assumptions. The proposed methodology, the achieved results, and the provided insights, other than being interesting on their own, when associated to the dataset that we made publicly available, will also pave the way for future research in the area.