Ultra-Wide-Field Scanning Laser Ophthalmoscopy (UWF-SLO) images capture high-resolution views of the retina with typically 200 spanning degrees. Accurate segmentation of vessels in UWF-SLO images is essential for detecting and diagnosing fundus disease. Recent studies have revealed that the selective State Space Model (SSM) in Mamba performs well in modeling long-range dependencies, which is crucial for capturing the continuity of elongated vessel structures. Inspired by this, we propose the first Serpentine Mamba (Serp-Mamba) network to address this challenging task. Specifically, we recognize the intricate, varied, and delicate nature of the tubular structure of vessels. Furthermore, the high-resolution of UWF-SLO images exacerbates the imbalance between the vessel and background categories. Based on the above observations, we first devise a Serpentine Interwoven Adaptive (SIA) scan mechanism, which scans UWF-SLO images along curved vessel structures in a snake-like crawling manner. This approach, consistent with vascular texture transformations, ensures the effective and continuous capture of curved vascular structure features. Second, we propose an Ambiguity-Driven Dual Recalibration (ADDR) module to address the category imbalance problem intensified by high-resolution images. Our ADDR module delineates pixels by two learnable thresholds and refines ambiguous pixels through a dual-driven strategy, thereby accurately distinguishing vessels and background regions. Experiment results on three datasets demonstrate the superior performance of our Serp-Mamba on high-resolution vessel segmentation. We also conduct a series of ablation studies to verify the impact of our designs. Our code shall be released upon publication of this work.