Adversarial training is one of the most effective defenses against adversarial attacks, but it incurs a high computational cost. In this study, we present the first theoretical analysis suggesting that adversarially pretrained transformers can serve as universally robust foundation models -- models that can adapt robustly to diverse downstream tasks with only lightweight tuning. Specifically, we demonstrate that single-layer linear transformers, after adversarial pretraining across a variety of classification tasks, can generalize robustly to unseen classification tasks through in-context learning from clean demonstrations (i.e., without requiring additional adversarial training or examples). This universal robustness stems from the model's ability to adaptively focus on robust features within given tasks. We also identify two open challenges for attaining robustness: the accuracy-robustness trade-off and sample-hungry training. This study initiates the discussion on the utility of universally robust foundation models. While their training is expensive, the investment would prove worthwhile as downstream tasks can obtain adversarial robustness for free. The code is available at https://github.com/s-kumano/universally-robust-in-context-learner.