The integration of gaze/eye tracking into virtual and augmented reality devices has unlocked new possibilities, offering a novel human-computer interaction (HCI) modality for on-device extended reality (XR). Emerging applications in XR, such as low-effort user authentication, mental health diagnosis, and foveated rendering, demand real-time eye tracking at high frequencies, a capability that current solutions struggle to deliver. To address this challenge, we present EX-Gaze, an event-based real-time eye tracking system designed for on-device extended reality. EX-Gaze achieves a high tracking frequency of 2KHz, providing decent accuracy and low tracking latency. The exceptional tracking frequency of EX-Gaze is achieved through the use of event cameras, cutting-edge, bio-inspired vision hardware that delivers event-stream output at high temporal resolution. We have developed a lightweight tracking framework that enables real-time pupil region localization and tracking on mobile devices. To effectively leverage the sparse nature of event-streams, we introduce the sparse event-patch representation and the corresponding sparse event patches transformer as key components to reduce computational time. Implemented on Jetson Orin Nano, a low-cost, small-sized mobile device with hybrid GPU and CPU components capable of parallel processing of multiple deep neural networks, EX-Gaze maximizes the computation power of Jetson Orin Nano through sophisticated computation scheduling and offloading between GPUs and CPUs. This enables EX-Gaze to achieve real-time tracking at 2KHz without accumulating latency. Evaluation on public datasets demonstrates that EX-Gaze outperforms other event-based eye tracking methods by striking the best balance between accuracy and efficiency on mobile devices. These results highlight EX-Gaze's potential as a groundbreaking technology to support XR applications that require high-frequency and real-time eye tracking. The code is available at https://github.com/Ningreka/EX-Gaze.