Um…sorta? A rudimentary AI simulation of them will live until power ceases, but that will be a separate entity than the flesh and blood gazillionaire who commissioned the AI to represent him.

On the other hand, it does intersect with the transporter paradox: When you beam down to the planet, is it the same you or just a clone with all your memories that is convincingly you even to itself? And if that’s the case, when you go into delta sleep and then wake again (at which point all your cognizant functions reboot) is the waking you the same person as the you who went to bed the night before?

If there’s any possibility that it isn’t then the gazillionaire AI zombie may be good enough.