The best example of human cells having programmed death is simply the main exception: offspring.
Children are made from a cell of the mother, combined with some DNA of the father. It is not created out of nothing or from scratch, it quite literally just one of the cells of the mother. So if a child survives one of the cells of the mother survives the mother's death.
Obviously the process involves resetting the "death counter". In fact there's a disease where that step seems to have gotten skipped: progeria.
Children are made from a cell of the mother, combined with some DNA of the father. It is not created out of nothing or from scratch, it quite literally just one of the cells of the mother. So if a child survives one of the cells of the mother survives the mother's death.
Obviously the process involves resetting the "death counter". In fact there's a disease where that step seems to have gotten skipped: progeria.