Complex hierarchies and branching patterns underlie numerous biological processes, from organismic development to signal divergence across individual cells. Single-cell RNA sequencing has enabled the study of such complex biological processes at high resolution. However, classical analysis methods represent single-cell data in low-dimensional Euclidean space, distorting the complex hierarchies inherent in such data. A recent line of work proposes to represent hierarchical data in hyperbolic space instead, which mitigates much of the distortion effects observed in the Euclidean setting. However, existing approaches for hyperbolic representation learning emphasise the preservation of local structure at the cost of decreased global representation accuracy and are computationally inefficient. To overcome these limitations, we develop Contrastive Poincaré Maps, a novel self-supervised approach for learning hyperbolic representations of tabular data. Through a series of experiments on synthetic and real single-cell data, we show that Contrastive Poincaré Maps accurately represent global structure in complex hierarchical data in a computationally efficient manner.