Lack of data is one of the biggest hurdles for rare disease research using deep learning. Due to the lack of rare-disease images and annotations, training a robust network for automatic rare-disease image segmentation is very challenging. To address this challenge, few-shot domain adaptation (FSDA) has emerged as a practical research direction, aiming to leverage a limited number of annotated images from a target domain to facilitate adaptation of models trained on other large datasets in a source domain. In this paper, we present a novel prototype-based feature mapping network (PFMNet) designed for FSDA in medical image segmentation. PFMNet adopts an encoder-decoder structure for segmentation, with the prototype-based feature mapping (PFM) module positioned at the bottom of the encoder-decoder structure. The PFM module transforms high-level features from the target domain into the source domain-like features that are more easily comprehensible by the decoder. By leveraging these source domain-like features, the decoder can effectively perform few-shot segmentation in the target domain and generate accurate segmentation masks. We evaluate the performance of PFMNet through experiments on three typical yet challenging few-shot medical image segmentation tasks: cross-center optic disc/cup segmentation, cross-center polyp segmentation, and cross-modality cardiac structure segmentation. We consider four different settings: 5-shot, 10-shot, 15-shot, and 20-shot. The experimental results substantiate the efficacy of our proposed approach for few-shot domain adaptation in medical image segmentation.