Ship tracking technology is crucial for emergency rescue in the event of a disaster. Quickly identifying the position and status of vessels is vital for rescue teams to be able to deploy efficiently in disaster areas. When responding to emergencies or natural disasters, ship tracking technology plays a critical role in supporting emergency rescue operations and resource allocation, improving the overall resilience of the maritime transportation system. However, the research on multi-object tracking (MOT) algorithms has primarily focused on optical image datasets. In contrast, image data from synthetic aperture radar (SAR) presents unique challenges, such as defocus interference, a high false alarm rate, and a lack of prior samples. To overcome these particular challenges, we propose a robust MOT algorithm developed for SAR images to achieve effective multi-vessel tracking under difficult imaging conditions. In particular, we optimize the YOLOv8 detection network by introducing a diffusion model-based training method for data augmentation. This method improves the robustness of the network to scaling, rotational and translational deformations. Moreover, an enhanced swin transformer is proposed as a feature extraction network, which strengthens the representation capability of the detection network. Furthermore, the state parameters within the KF technique are enhanced by directly capturing the details of the height and width of the tracking rectangle box. This refinement of the ByteTrack algorithm aims to achieve a more precise and accurate fit of the tracking rectangle to the ship, further improving the overall tracking performance. The experimental results from the ship detection and multiple objects tracking datasets show the impressive performance of the proposed model. With a precision of 97.60%, a recall of 96.36%, and an average precision of 96.72%, the model achieves exceptional detection accuracy with an 18% reduction in model parameters. Furthermore, significant improvements can be observed in key tracking metrics such as HOTA, MOTA and IDF1, with improvements of 4.8%, 8.5% and 6.8% respectively compared to the baseline algorithm, alongside a remarkable 37.5% reduction in IDS. It is noteworthy that the tracker works in real time, achieving an average analysis speed of 47 frames per second. The proposed MOT algorithm achieves state-of-art tracking performance on a SAR image dataset with short time sequences. Therefore, the proposed approach is a compelling solution for ship tracking in SAR imagery.