Both standalone language models (LMs) as well as LMs within downstream-task systems have been shown to generate statements which are factually untrue. This problem is especially severe for low-resource languages, where training data is scarce and of worse quality than for high-resource languages. In this opinion piece, we argue that LMs in their current state will never be fully trustworthy in critical settings and suggest a possible novel strategy to handle this issue: by building LMs such that can cite their sources - i.e., point a user to the parts of their training data that back up their outputs. We first discuss which current NLP tasks would or would not benefit from such models. We then highlight the expected benefits such models would bring, e.g., quick verifiability of statements. We end by outlining the individual tasks that would need to be solved on the way to developing LMs with the ability to cite. We hope to start a discussion about the field's current approach to building LMs, especially for low-resource languages, and the role of the training data in explaining model generations.