Firstly, it greatly depends on how you define immortality. If you define it as living forever and being indestructible as in a comic book, then, no, it is highly unlikely. However, if you define it in terms of showing no decline in survival characteristics, no increase in disease incidence and no increase in mortality with advancing age, then yes. The first is a science-fiction fantasy; the second is based on real-world biology that evolution has already selected for in certain species. We call this state negligible senescence.
Senescence and negligible senescence
Senescence refers to the gradual deterioration of aging and is typically very obvious in almost every species. More accurately, senescence refers to a decline of survival characteristics, such as strength, mobility, senses, and age-related increases in mortality along with a decrease in reproductive capability. Mortality rates for humans and most animals increase dramatically with age beyond reaching reproductive maturity.