Detecting flood-induced changes using synthetic aperture radar (SAR) is crucial for crisis management and damage assessment. Nevertheless, current methodologies predominantly focus on changes in buildings within optical images, struggling with the complex structures of floods. These structures are marked by widespread speckle noise and are accompanied by an increase in computational cost. These challenges hinder their success in real-world applications, necessitating a novel approach. This article proposes LiST-Net, a lightweight SAR transformer network with dimension-wise attention to improve flood detection accuracy. LiST-Net offers three key advantages. First, the graph neighbor module (GNM) is designed to enhance both detailed information of neighboring pixels and multidate features within the encoder. Second, the dimension-wise interactive attention (DIA) module is proposed to effectively reduce computational complexity while enhancing feature representation. Third, an attentive supervised learning module (ASLM) is incorporated to mitigate noise through a pixel mask gate, allowing change water information to pass through and improving the accuracy of water edge delineation. The effectiveness of LiST-Net is evaluated on two flood detection datasets, S1GFloods and ETCI-2021. Experimental results demonstrate that LiST-Net outperforms existing methods, showcasing a 94.7% improvement in F1 and an 88.7% enhancement in intersection over union (IoU) on the S1GFloods datasets, with lower computational costs (11.78G) and fewer parameters (7.34M). This underscores LiST-Net as a promising strategy for precise and effective mapping of floods within SAR images in real-world applications. A public release of the demo code will be available at https://github.com/Tamer-Saleh.