The use of polygonal and polyhedral finite element formulations has shown to be beneficial, especially if the structural geometry is complex or if local h-refinement is required. These elements must be applicable to a wide range of materials, including a class of hyperelastic materials. As a consequence, polygonal and polyhedral finite element methods must be extended to approximate problems at finite deformations. Here, we focus on the scaled boundary finite element method (SBFEM) for nonlinear analysis. The semi-analytical nature of the method allows for highly-accurate solutions in the scaling direction. Therefore, an extension of the SBFEM for nonlinear problems is introduced. To approximate the problem in a nonlinear setting, the variational formulation is linearized to accomodate a Newton-Raphson scheme. The scaled boundary finite element equation in displacement is set up for each increment. By expressing the incremental deformation gradient in a semi-analytical way, the residual is defined. The formulation reproduces the linear elastic solution after the first Newton-Raphson iteration. From the second iteration onward, the homogeneous Euler-Cauchy equation is transformed into a non-homogenous equation. It is shown that the semi-analytical approach fulfills the nonlinear patch test and is applicable to several benchmark problems.