The existence of noisy labels in real-world data negatively impacts the performance of deep learning models. Although much research effort has been devoted to improving robustness to noisy labels in classification tasks, the problem of noisy labels in deep metric learning (DML) remains open. In this paper, we propose a noise-resistant training technique for DML, which we name Probabilistic Ranking-based Instance Selection with Memory (PRISM). PRISM identifies noisy data in a minibatch using average similarity against image features extracted by several previous versions of the neural network. These features are stored in and retrieved from a memory bank. To alleviate the high computational cost brought by the memory bank, we introduce an acceleration method that replaces individual data points with the class centers. In extensive comparisons with 12 existing approaches under both synthetic and real-world label noise, PRISM demonstrates superior performance of up to 6.06% in Precision@1.