I believe we have immortal soul.
I don't think we could make our body immortal. We may be able to make our body live longer but not for infinity.
True, the soul is immortal, and the body can not live forever. We can extend only the amount of life, but not make it eternal. It is fantastic. Physically, the body will not withstand such a load, even to increase the life expectancy people will have to transplant organs and replace the skin, because they will wear out.