This study aimed to evaluate the biomechanics of single-leg drop landing in individuals with functional ankle instability (FAI) during cognitive tasks, contrasting these findings with those of healthy controls to provide insights for evidence-based rehabilitation strategies. Fifteen FAI participants, identified using clinical tools, were age- and activity-matched with controls. They performed drop landings with and without a cognitive task, and the data were analyzed using a 2 × 2 mixed ANOVA. At the initial ground contact (IC), the FAI group’s affected side showed a significantly smaller plantarflexion angle than the control group (p = 0.008). With cognitive tasks, this angle increased in the FAI group (p = 0.005). The FAI group also had larger knee flexion at contact (p = 0.002) and greater knee valgus at peak vertical ground reaction force (vGRF) (p = 0.027). They exhibited a higher peak vGRF, shorter time to peak vGRF (T-vGRF), and higher loading rate (LR) (all p < 0.05). No differences were found in other variables (p > 0.05). This study shows that FAI individuals make specific biomechanical adjustments under cognitive tasks, notably increased plantarflexion at IC, suggesting reactive compensations. Despite similar motor control to controls, this may reflect long-term adaptations rather than equal proficiency.