No there is not

The phoenix is part of mythology

No, since the phoenix never existed outside of mythology, it can't be extinct.

Flames, usually; the phoenix in mythology died and was reborn in fire.

The phoenix is a mythical, sacred firebird that can be found in Greek and Roman mythologies, though its roots are of Greek origin. The only other true association to other mythology is that its inspiration may have come from a similar Egyptian mythology creature called the Bennu. There are other firebirds from other mythologies, such as Persian, but that creature is known as the Simurgh, and it is a benevolent, mythical bird.