Lithium-ion batteries (LIBs) are currently the core energy storage component of electric vehicles (EVs). Real-time, accurate, and robust state-of-charge (SOC) estimation is critical for reliable range prediction and for preventing cell degradation due to overcharging or deep discharge. Current model-based estimation methods, based on equivalent circuit models and filters, such as the extended Kalman filter (EKF) and unscented Kalman filter (UKF), provide effective real-time SOC estimation. However, they typically assume a Gaussian process and measurement noise with known mean and covariance, which may not hold in practice. Distributionally robust Kalman filtering (DRKF) utilizes the Wasserstein distance to construct an ambiguity set around a nominal process and measurement noise, thereby accounting for errors in the nominal assumptions. In this paper, we propose the distributionally robust extended Kalman filter (DREKF), which broadens this methodology to nonlinear problems and applies it to the battery state estimation problem. We benchmark the DREKF against the EKF and UKF using simulations based on an open-source urban dynamometer driving schedule dataset. Results across multiple temperatures demonstrate that the DREKF achieves accuracy and computation time comparable to conventional EKF and UKF approaches under nominal conditions, while offering improved resilience to measurement outliers.