This paper describes the process of data processing and training of an automatic speech recognition (ASR) system for Cook Islands Māori (CIM), an Indigenous language spoken by approximately 22,000 people in the South Pacific. We transcribed four hours of speech from adults and elderly speakers of the language and prepared two experiments. First, we trained three ASR systems: one statistical, Kaldi; and two based on Deep Learning, DeepSpeech and XLSR-Wav2Vec2. Wav2Vec2 tied with Kaldi for lowest character error rate (CER=6±1) and was slightly behind in word error rate (WER=23±2 versus WER=18±2 for Kaldi). This provides evidence that Deep Learning ASR systems are reaching the performance of statistical methods on small datasets, and that they can work effectively with extremely low-resource Indigenous languages like CIM. In the second experiment we used Wav2Vec2 to train models with held-out speakers. While the performance decreased (CER=15±7, WER=46±16), the system still showed considerable learning. We intend to use ASR to accelerate the documentation of CIM, using newly transcribed texts to improve the ASR and also generate teaching and language revitalization materials. The trained model is available under a license based on the Kaitiakitanga License, which provides for non-commercial use while retaining control of the model by the Indigenous community.