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User-based KNN recommender systems (UserKNN) utilize the rating data of a target user's k nearest neighbors in the recommendation process. This, however, increases the privacy risk of the neighbors since their rating data might be exposed to other users or malicious parties. To reduce this risk, existing work applies differential privacy by adding randomness to the neighbors' ratings, which reduces the accuracy of UserKNN. In this work, we introduce ReuseKNN, a novel differentially-private KNN-based recommender system. The main idea is to identify small but highly reusable neighborhoods so that (i) only a minimal set of users requires protection with differential privacy, and (ii) most users do not need to be protected with differential privacy, since they are only rarely exploited as neighbors. In our experiments on five diverse datasets, we make two key observations: Firstly, ReuseKNN requires significantly smaller neighborhoods, and thus, fewer neighbors need to be protected with differential privacy compared to traditional UserKNN. Secondly, despite the small neighborhoods, ReuseKNN outperforms UserKNN and a fully differentially private approach in terms of accuracy. Overall, ReuseKNN leads to significantly less privacy risk for users than in the case of UserKNN.