What if proportional hazard assumption does not hold for a confounder in Cox PH model If proportionality hazard assumption does not appear to hold for a covariate in the Cox PH model, is it a serious matter?
The covariate is included in order to adjust for potential confounding. I am more interested in how the hazard ratio of the variable of interest will be affected if this assumption does not hold for the other covariates in the model.
Stratified Cox model might be a good approach to consider, but we do not have enough sample size.

If the covariate is very influential you sometimes benefit by adjusting for it as a covariate and also as a stratification factor.  Stratification will not hurt the effective sample size very much for estimating relative hazards, but will lower the precision of survival curve estimates.  But you need to state how you avoided linearity assumptions for continuous covariates.  Frequently, nonlinearities can be more impactful than non-proportional hazards.
Sometimes when you change models because of a failure to satisfy proportional hazards for one variable, you make the fit worse for other variables.  It can be helpful to get a sense of the overall "average" structure.  In RMS I have a detailed case study in a latter chapter where I examine the fit of a parametric survival model, and used an accelerated failure time model instead of the Cox model.
