Sensor fault diagnosis is of great significance to ensure safe battery operation. This paper proposes a novel sensor fault diagnosis method that achieves the simultaneous fault detection, fault source and type identification, and fault estimation in a comprehensive way. Specifically, a set-valued observer, featuring a state predictor and a state estimator, is first constructed and designed to guarantee the inclusion of the unavailable actual battery state due to unknown modeling errors and noises at every instant of time. Compared with the traditional observers, a distinct feature of the proposed one lies in that the calculated state predictions and estimations of the battery system at each time step are ellipsoidal sets in state space rather than single vectors. The boundedness of state prediction and estimation errors is formally proved, and the tractable design criteria for determining the real-time optimal prediction and estimation ellipsoids are also derived. As for diagnosis algorithm, fault detection is implemented based on the intersection between the prediction and estimation ellipsoids. Then, a two-layer Pearson correlation coefficient analysis mechanism is developed to identify the source and type of sensor faults. Another set-valued observer based on an augmented battery model is further designed to estimate the fault level. Finally, experimental studies of a battery cell under different sensor fault sources, types and values are elaborated to verify the effectiveness of the proposed method.