The Cox Proportional Hazards (CPH) model has long been the preferred survival model for its explainability. However, to increase its predictive power beyond its linear log-risk, it was extended to utilize deep neural networks, sacrificing its explainability. In this work, we explore the potential of self-explaining neural networks (SENN) for survival analysis. We propose a new locally explainable Cox proportional hazards model, named CoxSE, by estimating a locally-linear log-hazard function using the SENN. We also propose a modification to the Neural additive (NAM) model, hybrid with SENN, named CoxSENAM, which enables the control of the stability and consistency of the generated explanations. Several experiments using synthetic and real datasets are presented, benchmarking CoxSE and CoxSENAM against a NAM-based model, a DeepSurv model explained with SHAP, and a linear CPH model. The results show that, unlike the NAM-based model, the SENN-based model can provide more stable and consistent explanations while maintaining the predictive power of the black-box model. The results also show that, due to their structural design, NAM-based models demonstrate better robustness to non-informative features. Among the models, the hybrid model exhibits the best robustness.