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 mainly focused on terrestrial wireless devices while, to the best of our knowledge, none of the previous work considered 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 feature a low bit-rate and orbit at about 800 Km from the Earth, at a speed of around 25, 000 Km/h, thus making the receiver experiencing a down-link with unique attenuation and fading characteristics. In this paper, we investigate the effectiveness and main limitations of AI-based solutions to the physical-layer authentication of LEO satellites. Our study is performed on massive real data-more than 100M I-Q samplescollected from an extensive measurements campaign on the IRIDIUM LEO satellites constellation, lasting 589 hours. Our results show 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. However, the relatively high number of I-Q samples required by the proposed methodology, coupled with the low bandwidth of satellite link, might prevent the detection of the spoofing attack under certain configuration parameters.