For the design and fabrication of dies by the deep drawing process, the problems of springback after load removal and the formation of cracks must be considered. In this study, a deep drawing process for micro square holes on copper sheets was analyzed on the basis of an updated Lagrangian formulation and 3D finite element analysis. Sheet behavior was simulated using a micro-elastoplastic material model, the performance of which was compared with that of models involving conventional materials. Subsequently, Dynaform software was used for simulation analysis to obtain the material deformation history as well as determine the thickness change distribution and the maximum stress and strain of a copper sheet. It was found that the punch fillet radius and forming ratio affect the relationship between the punch load and the punch stroke, the distributions of stress and strain, and the maximum flange height. Finally, simulation results were compared with experimental results to confirm the accuracy of the 3D finite element analysis of the elastoplastic deformation. The results show the effect of the punch fillet radius for copper sheets on the drawing process: when the punch fillet radius is small, the punch load increases and decreases more rapidly. The maximum stress and strain decrease as the punch fillet radius increases. These findings serve as a valuable reference for design and processing by micro deep drawing.