An important risk that children face today is online grooming, where a so-called sexual predator establishes an emotional connection with a minor online with the objective of sexual abuse. Prior work has sought to automatically identify grooming chats, but only after an incidence has already happened in the context of legal prosecution. In this work, we instead investigate this problem from the point of view of prevention. We define and study the task of early sexual predator detection (eSPD) in chats, where the goal is to analyze a running chat from its beginning and predict grooming attempts as early and as accurately as possible. We survey existing datasets and their limitations regarding eSPD, and create a new dataset called PANC for more realistic evaluations. We present strong baselines built on BERT that also reach state-of-the-art results for conventional SPD. Finally, we consider coping with limited computational resources, as real-life applications require eSPD on mobile devices.