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The impact of the local chemical environment on the Gilbert damping in the binary alloy Fe100-xCox is investigated, using computations based on density functional theory. By varying the alloy composition x as well as Fe-Co atom positions we reveal that the effective damping of the alloy is highly sensitive to the nearest-neighbor environment, especially to the amount of Co and the average distance between Co-Co atoms at nearest-neighbor sites. Both lead to a significant local increase (up to an order of magnitude) of the effective Gilbert damping, originating mainly from variations of the density of states at the Fermi energy. In a global perspective (i.e., making a configuration average for a real material), those differences in damping are masked by statistical averages. When low-temperature explicit atomistic dynamics simulations are performed, the impact of short-range disorder on local dynamics is observed to also alter the overall relaxation rate. Our results illustrate the possibility of local chemical engineering of the Gilbert damping, which may stimulate the study of new ways to tune and control materials aiming for spintronics applications.