Immortality exists on mother earth but not in physical form. When spiritual travelers, seekers like Mahavira, Gautama Buddha even Jesus Christ and Prophet Mohammed indulging in spirituality succeeded in reaching state of enlightenment followed by moksha salvation, they became immortal forever through their teachings but never in physical form.
Immortality on mother earth can never be gained physically either in human form or lower forms of life. The state of immortality always belonged to spirit, prana soul atman existing in our heart.
Per se Hinduism scriptures a soul atman was indestructible, could neither be created or destroyed. Immortality is a property of soul atman from the beginning of time.
On mother earth a soul atman manifests human form to cleanse self of dross impurities within. The moment human beings reached stage of enlightenment and finally moksha salvation, the cosmic journey for soul atman finally ends.
The liberated soul atman being immortal can venture to any part of cosmos at will. Until it liberates, a soul atman remains bonded to solar system. Even before a soul atman starts its cosmic sojourn and after it liberates from endless cycles of birth and death, it was immortal.