This study presents the analytical formulation and the finite element solution of a geometrically nonlinear and nonlocal plate modeled via fractional-order mathematics. The finite nonlocal strains are obtained from a frame-invariant and thermodynamically consistent fractional-order continuum formulation. The fractional-order continuum model is used to capture softening nonlocal response via both Mindlin and Kirchhoff formulations. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal plates are obtained using variational principles. Further, a 2D nonlinear finite element model for the solution of this class of fractional-order systems is developed. The fractional finite element model is used to study the geometrically nonlinear response of nonlocal plates subject to various loading and boundary conditions. It is established that irrespective of the nature of boundary conditions, the fractional-order nonlocality leads to a reduction in the plate's stiffness thereby increasing the magnitude of the transverse displacements. More specifically, the nonlocal plate model based on fractional-order strain-displacement relations is free from boundary effects that lead to mathematical ill-posedness and inaccurate predictions typical of classical strain-driven integral approaches to nonlocal elasticity under certain loading and boundary conditions. This consistency in the predictions of the fractional-order nonlocal model is a result of the well-posed nature of the fractional-order governing equations. © 2020 Elsevier Ltd