In this paper, we present an approach to improve the robustness of BERT language models against word substitution-based adversarial attacks by leveraging adversarial perturbations for self-supervised contrastive learning. We create a word-level adversarial attack generating hard positives on-the-fly as adversarial examples during contrastive learning. In contrast to previous works, our method improves model robustness without using any labeled data. Experimental results show that our method improves robustness of BERT against four different word substitution-based adversarial attacks, and combining our method with adversarial training gives higher robustness than adversarial training alone. As our method improves the robustness of BERT purely with unlabeled data, it opens up the possibility of using large text datasets to train robust language models against word substitution-based adversarial attacks.