Will the Giant Beaver (Creature) always be available to tame?

Question

If I take too long to complete the quest to get the Colorful Water Lily item, will the Giant Beaver (Creature) always be available to tame?

Answer 1

Nope, as said by Mirade in Forum, you can only reach them during "beaver breakout" mini world change: 

You can only reach the beavers if a specific mini world change is active (Beaver Breakout). This happens with a certain probability at server save.

Source: Beaver Taming