This paper presents a neural machine translation model and dataset for the Chibchan language Bribri, with an average performance of BLEU 16.9±1.7. This was trained on an extremely small dataset (5923 Bribri-Spanish pairs), providing evidence for the applicability of NMT in extremely low-resource environments. We discuss the challenges entailed in managing training input from languages without standard orthographies, we provide evidence of successful learning of Bribri grammar, and also examine the translations of structures that are infrequent in major Indo-European languages, such as positional verbs, ergative markers, numerical classifiers and complex demonstrative systems. In addition to this, we perform an experiment of augmenting the dataset through iterative back-translation (Sennrich et al., 2016a; Hoang et al., 2018) by using Spanish sentences to create synthetic Bribri sentences. This improves the score by an average of 1.0 BLEU, but only when the new Spanish sentences belong to the same domain as the other Spanish examples. This contributes to the small but growing body of research on Chibchan NLP.