If it is a small proportion, then anything you do will have little impact on the results.
Most studies set an eligibility criterion, at design time, that patients can only be enrolled once. If that was not the case in your study, I suspect excluding observations after the first might introduce selection bias because people who died would not have had the chance to enter the study again after their death.
If you treat each observation as independent (ignoring the fact that some come from the same patient), your standard errors/confidence intervals/ and p-values will be too small. If you make some allowance for this “clustering” of observations within patients, you can avoid this. The trouble is that most of your “clusters” only have one observation and a few have ?4, so I’m not sure how the robust variance estimator that corrects the standard errors will perform. It might inflate them too much and therefore increase your p-values (I.e. increase risk of erroneously declaring “no statistical significance”).
I think reasonable to do anyway, though.
In any case, if the interpretation of your result is affected by these choices with so few affected patients, it is probably sensitive to many other issues too, including common things like confounding and somewhat nonproportional hazards.