Vascular segmentation in medical images is crucial for disease diagnosis and surgical navigation. However, the segmented vascular structure is often discontinuous due to its slender nature and inadequate prior modeling. In this paper, we propose a novel Serpentine Window Mamba (SWinMamba) to achieve accurate vascular segmentation. The proposed SWinMamba innovatively models the continuity of slender vascular structures by incorporating serpentine window sequences into bidirectional state space models. The serpentine window sequences enable efficient feature capturing by adaptively guiding global visual context modeling to the vascular structure. Specifically, the Serpentine Window Tokenizer (SWToken) adaptively splits the input image using overlapping serpentine window sequences, enabling flexible receptive fields (RFs) for vascular structure modeling. The Bidirectional Aggregation Module (BAM) integrates coherent local features in the RFs for vascular continuity representation. In addition, dual-domain learning with Spatial-Frequency Fusion Unit (SFFU) is designed to enhance the feature representation of vascular structure. Extensive experiments on three challenging datasets demonstrate that the proposed SWinMamba achieves superior performance with complete and connected vessels.