Recent progress in robotic manipulation has been fueled by large-scale datasets collected across diverse environments. Training robotic manipulation policies on these datasets is traditionally performed in a centralized manner, raising concerns regarding scalability, adaptability, and data privacy. While federated learning enables decentralized, privacy-preserving training, its application to robotic manipulation remains largely unexplored. We introduce FLAME (Federated Learning Across Manipulation Environments), the first benchmark designed for federated learning in robotic manipulation. FLAME consists of: (i) a set of large-scale datasets of over 160,000 expert demonstrations of multiple manipulation tasks, collected across a wide range of simulated environments; (ii) a training and evaluation framework for robotic policy learning in a federated setting. We evaluate standard federated learning algorithms in FLAME, showing their potential for distributed policy learning and highlighting key challenges. Our benchmark establishes a foundation for scalable, adaptive, and privacy-aware robotic learning. The code is publicly available at https://github.com/KTH-RPL/ELSA-Robotics-Challenge.