Objective: Accurate identification of ship maneuvering motion models is essential for motion prediction, heading control, trajectory tracking, and intelligent navigation. However, ship maneuvering responses are affected by hydrodynamic nonlinearities, rudder action, environmental disturbances, and measurement noise. Traditional response models, such as the Nomoto model, are physically interpretable, but their simplified structure limits their ability to describe complex free-running maneuvering data. Purely data-driven models, such as long short-term memory (LSTM) networks, can learn nonlinear time-series mappings but may exhibit limited generalization performance when the validation sequence differs from the training data. To improve physical interpretability and prediction accuracy, this paper proposes a physics-informed LSTM (PI-LSTM) method for system identification of ship maneuvering response models. Methods: A second-order nonlinear Nomoto response model was selected as the physical prior. The continuous Nomoto equation was discretized using finite-difference approximation, and the resulting discrete coefficients were directly identified as model parameters. A parametric physical branch was constructed to describe the ship's main yaw dynamics, whereas an LSTM network was introduced as a nonparametric residual branch to compensate for unmodeled dynamics, environmental disturbances, and measurement errors that cannot be explicitly represented by the simplified Nomoto model. The PI-LSTM prediction was expressed as the sum of the Nomoto physical prediction and the LSTM-based residual correction. The loss function consisted of a data-fitting term, a physical branch term, and a residual regularization term. An adaptive weighting strategy based on the exponential moving average was adopted to dynamically balance these terms during training. Free-running zigzag test data for the Esso Osaka scaled ship model under three conditions (30°/30°, 20°/20°, and 15°/15°) were used for training and validation. Given the temporal continuity of the data, each dataset was divided chronologically, with the first 70% used for training and the remaining 30% for validation. The standard LSTM, the identified Nomoto physical model, and the proposed PI-LSTM model were compared using the coefficient of determination (R²), root mean square error (RMSE), and symmetric mean absolute percentage error (SMAPE). Results: The identified discrete Nomoto parameters remained stable in magnitude across the zigzag conditions and followed physically reasonable trends. Parameters related to historical yaw-rate states were consistent across the three conditions, indicating that the main yaw dynamics were reliably identified. The nonlinear yaw-rate coefficient and the rudder-input-related coefficient varied with rudder angle amplitude, reflecting changes in maneuvering excitation intensity. The time-domain results showed that all three models could fit the training segment well. However, in the validation segment, the standard LSTM showed amplitude attenuation or trend deviation in some conditions, especially the 20°/20° and 15°/15° tests. The Nomoto model remained more stable owing to its physical structure, but its simplified form limited prediction accuracy. In comparison, the proposed PI-LSTM model better reproduced the yaw-rate peaks, sign changes, and heading-angle trends in both segments. Quantitatively, PI-LSTM generally achieved higher R² values and lower RMSE and SMAPE than the standard LSTM and the Nomoto model alone, especially in the validation segment. Conclusions: The proposed PI-LSTM method combines the physical interpretability of the second-order nonlinear Nomoto model with the nonlinear compensation ability of LSTM. It can simultaneously identify discrete maneuvering parameters and model nonparametric residual dynamics. Experiments on the Esso Osaka zigzag data show that the method improves the prediction accuracy and generalization performance of ship maneuvering response modeling, providing an effective approach for motion prediction, heading control, and intelligent navigation assistance.