The answer to all of the above is a clear no. Because you haven’t given away a part of yourself, you have given a part of your body. And your body is not you. Your soul is. You are a soul, and that soul has a body.
So then let’s ramp up the question a notch. Brain transplants are not yet possible. Although the way some people behave, it seems as if they have already become donors 🙂 But the theoretical possibility of brain transplants really boggles the mind. Am I still me if I have someone else’s brain? Maybe your kidney can become my kidney, and we both still remain the same people. But if someone’s brain becomes someone else’s, who is who?
Well, the answer remains the same. You are not your brain, you are your soul. Your mind, your personality and your identity are all stored in the soul, not the brain. Receiving someone else’s brain would be like buying a used smartphone. You just transfer your old information from the cloud to the new device, and life goes on.
So there is the answer. You are not a body, you are a soul. In the short time the soul inhabits the body, thank G-d for every healthy limb and organ you have, and for the opportunity to use them for good.
Menachem Mendel Bluming, Rabbi Moss and Chabad.org