The numerical solution of strain gradient elasticity problems is challenging due to higher-order derivatives and strong nonlocal effects, which can reduce the convergence of conventional methods such as the Finite Element Method (FEM). This work investigates the Physics-Informed Extreme Learning Machine (PIELM), a neural-network-based, mesh-free approach, as an alternative for accurately and efficiently solving these problems. Three benchmark cases with analytical solutions – static bending, free vibration, and linearized buckling of simply supported isotropic Kirchhoff nanoplates – are used to assess convergence and sensitivity to nonlocality. While FEM exhibits an algebraic convergence with errors increasing with the extent of nonlocality, PIELM achieves exponential convergence independently from the nonlocal scale.Training instabilities due to ill-conditioning can, however, limit the effective range of exponential error decay for large networks. Here, this issue is mitigated by enlarging the initialization of internal parameters. Overall, the results demonstrate that PIELM can efficiently approximate higherorder elastic fields and represents a promising, mesh-free alternative for the solution of nonlocal continuum models.