It is now clear that anthropogenic climate change is having a negative impact on human health. In this paper, we provide the first comprehensive assessment of the impact of climatic stressors on child health in Burkina Faso. We undertake a rigorous empirical analysis of the impact of climate and weather shocks on mortality, stunting (height‐for‐age Z‐score) and wasting (weight‐for‐age Z‐score), using Demographic and Health Surveys, combined with high‐resolution meteorological data, controlling for household and individual covariates. We find robust evidence that both lifetime and short‐term exposure to high temperatures and droughts have a negative impact on child health, as do increased temperature anomalies during crop seasons, suggesting a link between climate and health through domestic food production. Income and household wealth, access to electricity, sanitation and a health facility for childbirth negate some adverse impacts of climate change. Combining our econometric estimates with updated CMIP6 scenarios, we compute policy‐relevant projections of future child health. Our results show that future warming is projected to significantly increase child mortality, and share of underweight and stunted children, in all but the Paris Agreement scenario. Given the links between health, a key element of human capital, and economic growth, our findings and projections provide yet more evidence of the importance of a rapid reduction in global emissions combined with adaptation funding, if lower‐income countries are to achieve poverty reduction and increasing prosperity.