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In ecology it has become common to apply DNA barcoding to biological samples leading to datasets containing a large number of nucleotide sequences. The focus is then on inferring the taxonomic placement of each of these sequences by leveraging on existing databases containing reference sequences having known taxa. This is highly challenging because i) sequencing is typically only available for a relatively small region of the genome due to cost considerations; ii) many of the sequences are from organisms that are either unknown to science or for which there are no reference sequences available. These issues can lead to substantial classification uncertainty, particularly in inferring new taxa. To address these challenges, we propose a new class of Bayesian nonparametric taxonomic classifiers, BayesANT, which use species sampling model priors to allow new taxa to be discovered at each taxonomic rank. Using a simple product multinomial likelihood with conjugate Dirichlet priors at the lowest rank, a highly efficient algorithm is developed to provide a probabilistic prediction of the taxa placement of each sequence at each rank. BayesANT is shown to have excellent performance in real data, including when many sequences in the test set belong to taxa unobserved in training.