The fractional-order derivative Zener (FDZ) model can reasonably predict the frequency dependence behavior of rubber-like materials. However, its capability to capture nonlinear behaviors is limited. In our previous work, we built a horizontal shear model of a multidimensional rubber-like base isolation bearing, modifying the FDZ model to compensate for its inability to reproduce the nonlinear behaviors of amplitude dependence and slow stabilization. In this paper, the tension-compression behavior of the same bearing in the vertical direction is studied, where strain-stiffening nonlinear behavior is exhibited. A different strategy is used to establishing the modified tension-compression FDZ (CFDZ) model, which is shown to have good agreement with experimental results. State-space representation of the CFDZ model is presented and included in modelling a train-bridge-bearing system for dynamic analysis. Simulation results demonstrate that the proposed bearing achieves a high base isolation performance in the vertical direction.