Despite the tremendous success of deep neural networks in medical image segmentation, they typically require a large amount of costly, expert-level annotated data. Few-shot segmentation approaches address this issue by learning to transfer knowledge from limited quantities of labeled examples. Incorporating appropriate prior knowledge is critical in designing high-performance few-shot segmentation algorithms. Since strong spatial priors exist in many medical imaging modalities, we propose a prototype-based method -- namely, the location-sensitive local prototype network -- that leverages spatial priors to perform few-shot medical image segmentation. Our approach divides the difficult problem of segmenting the entire image with global prototypes into easily solvable subproblems of local region segmentation with local prototypes. For organ segmentation experiments on the VISCERAL CT image dataset, our method outperforms the state-of-the-art approaches by 10% in the mean Dice coefficient. Extensive ablation studies demonstrate the substantial benefits of incorporating spatial information and confirm the effectiveness of our approach.