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From the mitotic spindle up to tissues and biofilms, many biological systems behave as active droplets, which often break symmetry and change shape spontaneously. Here, I show that active nematic droplets can experience a fingering instability. I consider an active fluid that acquires nematic order through anchoring at the droplet interface, and I predict its morphological stability in terms of three dimensionless parameters: the anchoring angle, the penetration length of nematic order compared to droplet size, and an active capillary number. Droplets with extensile (contractile) stresses and planar (homeotropic) anchoring are unstable above a critical activity or droplet size. This instability is interfacial in nature: It arises through the coupling of active flows with interface motion, even when the bulk instability of active nematics cannot take place. In contrast to the dynamic states characteristic of active matter, the instability could produce static fingering patterns. The number of fingers increases with activity but varies non-monotonically with the nematic penetration length. Overall, these results can help to understand the self-organized shapes of biological systems, and to design patterns in active materials.