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Data drift is a thorny challenge when deploying person re-identification (ReID) models into real-world devices, where the data distribution is significantly different from that of the training environment and keeps changing. To tackle this issue, we propose a federated spatial-temporal incremental learning approach, named FedSTIL, which leverages both lifelong learning and federated learning to continuously optimize models deployed on many distributed edge clients. Unlike previous efforts, FedSTIL aims to mine spatial-temporal correlations among the knowledge learnt from different edge clients. Specifically, the edge clients first periodically extract general representations of drifted data to optimize their local models. Then, the learnt knowledge from edge clients will be aggregated by centralized parameter server, where the knowledge will be selectively and attentively distilled from spatial- and temporal-dimension with carefully designed mechanisms. Finally, the distilled informative spatial-temporal knowledge will be sent back to correlated edge clients to further improve the recognition accuracy of each edge client with a lifelong learning method. Extensive experiments on a mixture of five real-world datasets demonstrate that our method outperforms others by nearly 4% in Rank-1 accuracy, while reducing communication cost by 62%. All implementation codes are publicly available on https://github.com/MSNLAB/Federated-Lifelong-Person-ReID