This paper introduces Low-shot Object Learning with Mutual Exclusivity Bias (LSME), the first computational framing of mutual exclusivity bias, a phenomenon commonly observed in infants during word learning. We provide a novel dataset, comprehensive baselines, and a SOTA method to enable the ML community to tackle this challenging learning task. The goal of LSME is to analyze an RGB image of a scene containing multiple objects and correctly associate a previously-unknown object instance with a provided category label. This association is then used to perform low-shot learning to test category generalization. We provide a data generation pipeline for the LSME problem and conduct a thorough analysis of the factors that contribute to its difficulty. Additionally, we evaluate the performance of multiple baselines, including state-of-the-art foundation models. Finally, we present a baseline approach that outperforms state-of-the-art models in terms of low-shot accuracy. Code and data are available at https://github.com/rehg-lab/LSME.