Clustering is a ubiquitous task in data science. Compared to the commonly used $k$-means clustering, $k$-medoids clustering requires the cluster centers to be actual data points and support arbitrary distance metrics, which permits greater interpretability and the clustering of structured objects. Current state-of-the-art $k$-medoids clustering algorithms, such as Partitioning Around Medoids (PAM), are iterative and are quadratic in the dataset size $n$ for each iteration, being prohibitively expensive for large datasets. We propose BanditPAM, a randomized algorithm inspired by techniques from multi-armed bandits, that reduces the complexity of each PAM iteration from $O(n^2)$ to $O(n \log n)$ and returns the same results with high probability, under assumptions on the data that often hold in practice. As such, BanditPAM matches state-of-the-art clustering loss while reaching solutions much faster. We empirically validate our results on several large real-world datasets, including a coding exercise submissions dataset, the 10x Genomics 68k PBMC single-cell RNA sequencing dataset, and the MNIST handwritten digits dataset. In these experiments, we observe that BanditPAM returns the same results as state-of-the-art PAM-like algorithms up to 4x faster while performing up to 200x fewer distance computations. The improvements demonstrated by BanditPAM enable $k$-medoids clustering on a wide range of applications, including identifying cell types in large-scale single-cell data and providing scalable feedback for students learning computer science online. We also release highly optimized Python and C++ implementations of our algorithm.