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Sensor errors, model uncertainty, changes in the surrounding environment, data loss, or malicious network attacks can all result in outliers, which can pollute the measurement process of the distribution system. In this study, to address this problem, a dynamic state estimation method for distribution systems based on innovation-saturated Koopman Kalman filter (IS-KKF) was proposed. The Koopman Kalman filter (KKF) is model-free and data-driven, and estimates nonlinear dynamics through the conventional linear Kalman filter. To make the Koopman Kalman filter robust to measurement outliers, the innovation saturation mechanism applies a saturation function to it. To correct the state estimation, this mechanism is applied to the filtering process. When outliers occur, the distorted innovation is saturated to prevent the state estimation results from being destroyed. Adaptive adjustment of the saturated boundary is a feature of this mechanism. Extensive simulations were carried out on the IEEE 118-bus test system to verify the effectiveness and robustness of the proposed method. The simulation results demonstrate that the proposed method can effectively reject outliers with different amplitudes, types, and continuous times, has significant computational efficiency, and does not require additional measurement redundancy.