We lose our identity only if we lose our memory. Otherwise we are propelled along, tied to the underside of our "self"-imposed story. We are drawn, quartered & reassembled every second.
We lose our identity only if we lose our memory. Yet the people who know us carry a part of us with them, a copy of a copy: their memory of who they perceived us to be. Our memory is gone, but their memories of us create a kind of a back-up.
We lose our identity only if we lose our memory. If some recollections remain, we may fill in the gaps ourselves, but others may reject our extrapolation. In the previous scenario, our loved ones are free to imagine us as they remember, but if we are still alive, and act differently than they remember, then we will appear to them as the hauntings of a corpse.