One-shot medical image segmentation (MIS) is crucial for medical analysis due to the burden of medical experts on manual annotation. The recent emergence of the segment anything model (SAM) has demonstrated remarkable adaptation in MIS but cannot be directly applied to one-shot medical image segmentation (MIS) due to its reliance on labor-intensive user interactions and the high computational cost. To cope with these limitations, we propose a novel SAM-guided robust representation learning framework, named RRL-MedSAM, to adapt SAM to one-shot 3D MIS, which exploits the strong generalization capabilities of the SAM encoder to learn better feature representation. We devise a dual-stage knowledge distillation (DSKD) strategy to distill general knowledge between natural and medical images from the foundation model to train a lightweight encoder, and then adopt a mutual exponential moving average (mutual-EMA) to update the weights of the general lightweight encoder and medical-specific encoder. Specifically, pseudo labels from the registration network are used to perform mutual supervision for such two encoders. Moreover, we introduce an auto-prompting (AP) segmentation decoder which adopts the mask generated from the general lightweight model as a prompt to assist the medical-specific model in boosting the final segmentation performance. Extensive experiments conducted on three public datasets, i.e., OASIS, CT-lung demonstrate that the proposed RRL-MedSAM outperforms state-of-the-art one-shot MIS methods for both segmentation and registration tasks. Especially, our lightweight encoder uses only 3\% of the parameters compared to the encoder of SAM-Base.