Event-based eye tracking has shown great promise with the high temporal resolution and low redundancy provided by the event camera. However, the diversity and abruptness of eye movement patterns, including blinking, fixating, saccades, and smooth pursuit, pose significant challenges for eye localization. To achieve a stable event-based eye-tracking system, this paper proposes a bidirectional long-term sequence modeling and time-varying state selection mechanism to fully utilize contextual temporal information in response to the variability of eye movements. Specifically, the MambaPupil network is proposed, which consists of the multi-layer convolutional encoder to extract features from the event representations, a bidirectional Gated Recurrent Unit (GRU), and a Linear Time-Varying State Space Module (LTV-SSM), to selectively capture contextual correlation from the forward and backward temporal relationship. Furthermore, the Bina-rep is utilized as a compact event representation, and the tailor-made data augmentation, called as Event-Cutout, is proposed to enhance the model's robustness by applying spatial random masking to the event image. The evaluation on the ThreeET-plus benchmark shows the superior performance of the MambaPupil, which secured the 1st place in CVPR'2024 AIS Event-based Eye Tracking challenge.