Medical image segmentation typically demands extensive dense annotations for model training, which is both time-consuming and skill-intensive. To mitigate this burden, exemplar-based medical image segmentation methods have been introduced to achieve effective training with only one annotated image. In this paper, we introduce a novel Cross-model Mutual learning framework for Exemplar-based Medical image Segmentation (CMEMS), which leverages two models to mutually excavate implicit information from unlabeled data at multiple granularities. CMEMS can eliminate confirmation bias and enable collaborative training to learn complementary information by enforcing consistency at different granularities across models. Concretely, cross-model image perturbation based mutual learning is devised by using weakly perturbed images to generate high-confidence pseudo-labels, supervising predictions of strongly perturbed images across models. This approach enables joint pursuit of prediction consistency at the image granularity. Moreover, cross-model multi-level feature perturbation based mutual learning is designed by letting pseudo-labels supervise predictions from perturbed multi-level features with different resolutions, which can broaden the perturbation space and enhance the robustness of our framework. CMEMS is jointly trained using exemplar data, synthetic data, and unlabeled data in an end-to-end manner. Experimental results on two medical image datasets indicate that the proposed CMEMS outperforms the state-of-the-art segmentation methods with extremely limited supervision.