Is it okay to say : Some of the animals have become extinguished?

The word that you probably have in mind is extinct, not extinguished. When something becomes extinct, it ceases to exist. You don’t’ find it anywhere on the planet. Any living organism can become extinct.

Dinosaurs, for example, have become extinct.

There are reasons to believe that the tiger will soon become extinct.

Some of the animals in the rainforests are on the verge of extinction.

The word extinguish on the other hand is usually used along with the word Fire. When you extinguish a fire or a flame, you put it out.

The firemen battled for three hours before they extinguished the fire.