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We aim at the problem named One-Shot Unsupervised Domain Adaptation. Unlike traditional Unsupervised Domain Adaptation, it assumes that only one unlabeled target sample can be available when learning to adapt. This setting is realistic but more challenging, in which conventional adaptation approaches are prone to failure due to the scarce of unlabeled target data. To this end, we propose a novel Adversarial Style Mining approach, which combines the style transfer module and task-specific module into an adversarial manner. Specifically, the style transfer module iteratively searches for harder stylized images around the one-shot target sample according to the current learning state, leading the task model to explore the potential styles that are difficult to solve in the almost unseen target domain, thus boosting the adaptation performance in a data-scarce scenario. The adversarial learning framework makes the style transfer module and task-specific module benefit each other during the competition. Extensive experiments on both cross-domain classification and segmentation benchmarks verify that ASM achieves state-of-the-art adaptation performance under the challenging one-shot setting.