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The Pirahã language has been at the center of recent debates in linguistics, in large part because it is claimed not to exhibit recursion, a purported universal of human language. Here, we present an analysis of a novel corpus of natural Pirahã speech that was originally collected by Dan Everett and Steve Sheldon. We make the corpus freely available for further research. In the corpus, Pirahã sentences have been shallowly parsed and given morpheme-aligned English translations. We use the corpus to investigate the formal complexity of Pirahã syntax by searching for evidence of syntactic embedding. In particular, we search for sentences which could be analyzed as containing center-embedding, sentential complements, adverbials, complementizers, embedded possessors, conjunction or disjunction. We do not find unambiguous evidence for recursive embedding of sentences or noun phrases in the corpus. We find that the corpus is plausibly consistent with an analysis of Pirahã as a regular language, although this is not the only plausible analysis.