It is possible to explore higher dimensional topological properties in lower dimensional structures by introducing additional synthetic dimensions. In this Letter, we construct a four-dimensional (4D) second-order topological insulator using gradient nanoparticle arrays arranged in a periodic lattice. The nanoparticle array has spatially varying inter-particle distance along x and y directions, which can be regarded as two synthetic dimensions. Different from higher-order topological insulators in classical wave systems, the higher-order topological states in this 4D system are protected by a pair of first Chern numbers in two-dimensional (2D) subspaces instead of by the quantized 2D Zak phases. It is shown that there exist (4-1)- and (4-2)-dimensional boundary states for both transverse and longitudinal collective resonant modes, which provides new, to the best of our knowledge, mechanisms for light confinement and control in such a plasmonic superlattice.