Recent work has shown that, while large language models (LLMs) demonstrate strong word translation or bilingual lexicon induction (BLI) capabilities in few-shot setups, they still cannot match the performance of 'traditional' mapping-based approaches in the unsupervised scenario where no seed translation pairs are available, especially for lower-resource languages. To address this challenge with LLMs, we propose self-augmented in-context learning (SAIL) for unsupervised BLI: starting from a zero-shot prompt, SAIL iteratively induces a set of high-confidence word translation pairs for in-context learning (ICL) from an LLM, which it then reapplies to the same LLM in the ICL fashion. Our method shows substantial gains over zero-shot prompting of LLMs on two established BLI benchmarks spanning a wide range of language pairs, also outperforming mapping-based baselines across the board. In addition to achieving state-of-the-art unsupervised BLI performance, we also conduct comprehensive analyses on SAIL and discuss its limitations.