The oscillatory dynamics in droplets of the ferroin-catalyzed Belousov-Zhabotinsky solution is investigated. Due to a diffusive influx of molecular oxygen from a surrounding hydrophobic phase, droplets experience an increase in oscillation period with decreasing drop radius. If the radius is smaller than approximately 1.5 mm, no oscillations are observed. The phase of oscillations showed a radial dependence that gives rise to nontransient phase waves propagating from the center outward. In the absence of oxygen, the period decreases slightly with decreasing drop radii. A simple geometrical analysis qualitatively explains the main experimental observations. The analysis stems from an approach that relates the efficiency of inhibition to the surface-to-volume ratio of droplets.