Extreme climate events can have important consequences for the dynamics of natural populations, and severe droughts are predicted to become more common and intense due to climate change. We analysed infant mortality in relation to drought in two primate species (white-faced capuchins, Cebus capucinus imitator, and Geoffroy's spider monkeys, Ateles geoffroyi) in a tropical dry forest in north-western Costa Rica. Our survival analyses combine several rare and valuable long-term data sets, including long-term primate life-history, landscape-scale fruit abundance, food-tree mortality, and climate conditions. Infant capuchins showed a threshold mortality response to drought, with exceptionally high mortality during a period of intense drought, but not during periods of moderate water shortage. In contrast, spider monkey females stopped reproducing during severe drought, and the mortality of infant spider monkeys peaked later during a period of low fruit abundance and high food-tree mortality linked to the drought. These divergent patterns implicate differing physiology, behaviour, or associated factors in shaping species-specific drought responses. Our findings link predictions about the Earth’s changing climate to environmental influences on primate mortality risk and thereby improve our understanding of how the increasing severity and frequency of droughts will affect the dynamics and conservation of wild primates.