Background and Purpose: Functional assessment of the left ventricle using gated myocardial perfusion (MPS) single-photon emission computed tomography relies on the precise extraction of the left ventricular contours while simultaneously ensuring the security of patient data. Methods: In this paper, we introduce the integration of Federated Domain Adaptation with TimeSformer, named 'FedDA-TSformer' for left ventricle segmentation using MPS. FedDA-TSformer captures spatial and temporal features in gated MPS images, leveraging spatial attention, temporal attention, and federated learning for improved domain adaptation while ensuring patient data security. In detail, we employed Divide-Space-Time-Attention mechanism to extract spatio-temporal correlations from the multi-centered MPS datasets, ensuring that predictions are spatio-temporally consistent. To achieve domain adaptation, we align the model output on MPS from three different centers using local maximum mean discrepancy (LMMD) loss. This approach effectively addresses the dual requirements of federated learning and domain adaptation, enhancing the model's performance during training with multi-site datasets while ensuring the protection of data from different hospitals. Results: Our FedDA-TSformer was trained and evaluated using MPS datasets collected from three hospitals, comprising a total of 150 subjects. Each subject's cardiac cycle was divided into eight gates. The model achieved Dice Similarity Coefficients (DSC) of 0.842 and 0.907 for left ventricular (LV) endocardium and epicardium segmentation, respectively. Conclusion: Our proposed FedDA-TSformer model addresses the challenge of multi-center generalization, ensures patient data privacy protection, and demonstrates effectiveness in left ventricular (LV) segmentation.