Recently, Depth Anything Models (DAMs) - a type of depth foundation models - have demonstrated impressive zero-shot capabilities across diverse perspective images. Despite its success, it remains an open question regarding DAMs' performance on panorama images that enjoy a large field-of-view (180x360) but suffer from spherical distortions. To address this gap, we conduct an empirical analysis to evaluate the performance of DAMs on panoramic images and identify their limitations. For this, we undertake comprehensive experiments to assess the performance of DAMs from three key factors: panoramic representations, 360 camera positions for capturing scenarios, and spherical spatial transformations. This way, we reveal some key findings, e.g., DAMs are sensitive to spatial transformations. We then propose a semi-supervised learning (SSL) framework to learn a panoramic DAM, dubbed PanDA. Under the umbrella of SSL, PanDA first learns a teacher model by fine-tuning DAM through joint training on synthetic indoor and outdoor panoramic datasets. Then, a student model is trained using large-scale unlabeled data, leveraging pseudo-labels generated by the teacher model. To enhance PanDA's generalization capability, M\"obius transformation-based spatial augmentation (MTSA) is proposed to impose consistency regularization between the predicted depth maps from the original and spatially transformed ones. This subtly improves the student model's robustness to various spatial transformations, even under severe distortions. Extensive experiments demonstrate that PanDA exhibits remarkable zero-shot capability across diverse scenes, and outperforms the data-specific panoramic depth estimation methods on two popular real-world benchmarks.