Saliency-aware portion-wise mixup has proven to be an effective data augmentation technique for different modalities and tasks. However, it involves calculating the saliency over gradient vectors in the Euclidean space, representations that often possess complicated geometries and inherent hierarchical structure. We propose PISA saliency-aware interpolative regularization operating in the hyperbolic space, to better capture the complex geometries of representations. To this end, we also formulate a saliency-aware mixup for speech signals. PISA outperforms existing state-of-the-art interpolative augmentation methods on 7 benchmark and low-resource datasets from the domains of speech signal processing and computer vision. PISA results in more stable training than existing data augmentation methods while being robust to adversarial attacks. It can be generalized across modalities, models and downstream tasks.