After a few hundereed year if someone found any DNA belonging to you it is possible that you could be cloned so you can never be dead.
As your cells weren’t alive and multiplying in the mean time, you could as well claim to have been dead, though. (Think of the tax deductions!)
HeLa cells have been alive all the time since the 50’s though, and probably will be for some time. So does this mean that Henrietta Lacks is not dead yet?
You are not your DNA, or even your cells. Those provide the instructions and comprise they physical instantiation of you, but from any practical sense, you are the continuity of memories and self-awareness that forms your basic personality and relationships with people you know. This is why we widely recognize “brain death” (e.g. irreversible damage and loss of cognitive function) as being equivalent to the death of a person as a sentient entity.
Stranger
I’ve heard that your hair and nails keep growing even after death.
While perfectly true, this is not what the OP asked:
This overlooks the possibility of new cells being produced after the heart has stopped. (I can’t imagine that failure of the heart would cause all cellular meiosis and mitosis elsewhere in the body to instantly cease.)
That’s actually a myth. It can appear that way because as the skin becomes desiccated, it retracts a bit and so reveals more of the hair and nails that were already there.