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This study presents the framework to perform a stability analysis of nonlocal solids whose response is formulated according to the fractional-order continuum theory. In this formulation, space fractional-order operators are used to capture the nonlocal response of the medium by introducing nonlocal kinematic relations. First, we use the geometrically nonlinear fractional-order kinematic relations within an energy-based approach to establish the Lagrange-Dirichlet stability criteria for fractional-order nonlocal structures. This energy-based approach to nonlocal structural stability is possible due to a positive-definite and thermodynamically consistent definition of deformation energy enabled by the fractional-order kinematic formulation. Then, the Rayleigh-Ritz coefficient for the critical load is derived for linear buckling conditions. The fractional-order formulation is finally used to determine critical buckling loads of slender nonlocal beams and plates using a dedicated fractional-order finite element solver. Results establish that, in contrast to existing studies, the effect of nonlocal interactions is observed on both the material and the geometric stiffness, when using the fractional-order kinematics approach. We support these observations quantitatively with the help of case studies focusing on the critical buckling response of fractional-order nonlocal slender structures, and qualitatively via direct comparison of the fractional-order approach with the classical nonlocal approaches.