Medical image segmentation often faces the challenge of prohibitively expensive annotation costs. While few-shot learning offers a promising solution to alleviate this burden, conventional approaches still rely heavily on pre-training with large volumes of labeled data from known categories. To address this issue, we propose leveraging the Segment Anything Model (SAM), pre-trained on over 1 billion masks, thus circumventing the need for extensive domain-specific annotated data. In light of this, we developed SAM-MPA, an innovative SAM-based framework for few-shot medical image segmentation using Mask Propagation-based Auto-prompting. Initially, we employ k-centroid clustering to select the most representative examples for labelling to construct the support set. These annotated examples are registered to other images yielding deformation fields that facilitate the propagation of the mask knowledge to obtain coarse masks across the dataset. Subsequently, we automatically generate visual prompts based on the region and boundary expansion of the coarse mask, including points, box and a coarse mask. Finally, we can obtain the segmentation predictions by inputting these prompts into SAM and refine the results by post refinement module. We validate the performance of the proposed framework through extensive experiments conducted on two medical image datasets with different modalities. Our method achieves Dices of 74.53%, 94.36% on Breast US, Chest X-ray, respectively. Experimental results substantiate that SAM-MPA yields high-accuracy segmentations within 10 labeled examples, outperforming other state-of-the-art few-shot auto-segmentation methods. Our method enables the customization of SAM for any medical image dataset with a small number of labeled examples.