In the paper, the Lagrange geometric interpolation by spatial rational cubic Bézier curves is studied. It is shown that under some natural conditions the solution of the interpolation problem exists and is unique. Furthermore, it is given in a simple closed form which makes it attractive for practical applications. Asymptotic analysis confirms the expected approximation order, i.e., order six. Numerical examples pave the way for a promising nonlinear geometric subdivision scheme. ► Lagrange geometric interpolation of spatial data by rational cubic curves. ► Analysis of the solvability conditions and the solution given in a closed form. ► Optimal approximation order established. ► Nonlinear subdivision based on the developed Lagrange scheme derived.