The occurrence of hydrological drought, caused by rainfall deficiency, poses a threat to forest areas—not only due to the danger of fire, but as a result of changes in habitat conditions. It is predicted that more frequent periods of drought and high temperatures will reduce the stability of forests and increase their susceptibility to industrial pollution, pressure from insect pests, fungal pathogens, and fires. The main aim of this study was to investigate the current streamflow droughts in the catchment area of the Narewka River as an indicator of hydrological droughts in the Białowieża Primeval Forest for the period 1951–2020. The research presented in this paper shows that low flows of surface waters have appeared as an indicator of hydrological droughts almost every year since 1983. The analysis of the trend over the past 70 years shows that the aggregated annual volumes of low flows and the number of days with low flows are increasing significantly. This indicates a significant and relatively permanent change in the seasonal structure of the river runoff in the event of low summer flow.