This paper explores the use of hyperbolic geometry and deep learning techniques for recommendation. We present Hyperbolic Neural Collaborative Recommender (HNCR), a deep hyperbolic representation learning method that exploits mutual semantic relations among users/items for collaborative filtering (CF) tasks. HNCR contains two major phases: neighbor construction and recommendation framework. The first phase introduces a neighbor construction strategy to construct a semantic neighbor set for each user and item according to the user-item historical interaction. In the second phase, we develop a deep framework based on hyperbolic geometry to integrate constructed neighbor sets into recommendation. Via a series of extensive experiments, we show that HNCR outperforms its Euclidean counterpart and state-of-the-art baselines.