In high-resolution automotive radar tracking systems, vehicle targets are often regarded as extended targets, which means multiple measurements originated from scattering centers of vehicle targets can be detected at each scan and thus the traditional point target tracking schemes are unsuitable. Meanwhile, vehicle maneuvers, e.g., braking and swerving, cause serious degradation of the classical extended target tracking methods. In this paper, a novel method is proposed for maneuvering extended vehicle tracking with automotive radar. The data-region association (DRA) strategy is adopted to handle the vehicle extension effect, which is superior in describing the complex spatial distribution of vehicle target measurements. The interacting multiple model (IMM) method is combined with this DRA strategy to describe the evolution of target motion models. Accordingly, the proposed DRA-IMM method achieves satisfying tracking performance of extended vehicles and also guarantees the robustness in case of maneuvers. Furthermore, in view of the correlation between vehicle extension and its kinematic state, a ray-based strategy is devised to improve the prior distribution of the data-region association of the basic DRA-IMM, and accordingly an enhanced DRA-IMM (EDRA-IMM) method is proposed. Simulation result validates the effectiveness of the proposed DRA-IMM method for maneuvering extended vehicle tracking and the further improvement of the proposed EDRA-IMM method.