Almost all Akatsuki members showed their immortality. For example:

1. Sasori’s immortality lies on his puppets. Sasori has 100 puppets that look like his own original body puppet. These puppets will become Sasori himself when the source of Sasori’s chakra (heart) will be fixed into a puppet.
2. Deidara has the ability to use someones body parts as if they are his once the said part will be attached to his body.
3. Kakuzu has the ability to use someones heart and made those hearts his backup once the heart he used will be killed.
4. Hidan can live even if he has no body as long as his brain lives. And if his head will be attached to someones body, the body will become Hidan’s.
5. Orochimaru showed his immortality by transferring his soul to someones body.
6. Pain, the leader of the Akatsuki became immortal in the sense that he has several bodies. As of writing this post, it was not yet clear how he manages these bodies.

The immortality of the other members of the Akatsuki is yet revealed on the Naruto story. But what made me wonder is the immortality of Uchiha Itachi. Uchiha Itachi became one of the main characters of the Naruto story since from the start of the Naruto he was already mentioned as the brother of Sasuke that caused problem in the relationship between Naruto and Sasuke, and Sasuke to the whole village of Konoha.