In this paper, the recently-developed floating node method is extended for damage analysis of laminated composites with large deformations. Strong discontinuities including interfacial delamination and matrix cracks are explicitly represented by geometrically nonlinear kinematics. Interactions between these two kinds of failure patterns are enabled through enriched elements equipped with floating nodes. A cohesive zone model is utilized for the damage process zone. A general implicit procedure with user-defined elements is developed for both quasi-static and dynamic analysis. The performance of this formulation is verified with two benchmark simulations, involving buckling-induced delamination and low-velocity impact damage. The results presented show good quantitative and qualitative agreements with results from literature.