Learning to rank is an important problem in machine learning and recommender systems. In a recommender system, a user is typically recommended a list of items. Since the user is unlikely to examine the entire recommended list, partial feedback arises naturally. At the same time, diverse recommendations are important because it is challenging to model all tastes of the user in practice. In this paper, we propose the first algorithm for online learning to rank diverse items from partial-click feedback. We assume that the user examines the list of recommended items until the user is attracted by an item, which is clicked, and does not examine the rest of the items. This model of user behavior is known as the cascade model. We propose an online learning algorithm, cascadelsb, for solving our problem. The algorithm actively explores the tastes of the user with the objective of learning to recommend the optimal diverse list. We analyze the algorithm and prove a gap-free upper bound on its n-step regret. We evaluate cascadelsb on both synthetic and real-world datasets, compare it to various baselines, and show that it learns even when our modeling assumptions do not hold exactly.