Sodium montmorillonite (Na-MMT) clay mineral is a common type of swelling clay that has potential applications for nuclear waste storage at high temperatures and pressures. However, there is a limited understanding of the mechanical properties, local molecular stiffness, and dynamic heterogeneity of this material at elevated temperatures and pressures. To address this, we employ all-atomistic (AA) molecular dynamics (MD) simulation to investigate the tensile behavior of Na-MMT clay over a wide temperature range (500 K to 1700 K) and pressures (200 atm to 100 000 atm). The results show that increasing the temperature significantly reduces the tensile modulus, strength, and failure strain, while pressure has a minor effect compared to temperature, as seen in the normalized pressure-temperature plot. Mean-square displacement (MSD) analysis reveals increased molecular stiffness with increasing pressure and decreasing temperature, indicating suppressed atomic mobility. Our simulations indicate temperature-dependent dynamical heterogeneity in the Na-MMT model, supported by experimental studies and quantified local molecular stiffness distribution. These findings enhance our understanding of the tensile response and dynamical heterogeneity of Na-MMT clay under extreme conditions, aiding the development of clay minerals for engineering applications such as nuclear waste storage and shale gas extraction.