To get a god chance to knock it out and get it out of the dangerous areas they spawn in, make a "trap boat" and lure it in one side, walk out a door in the other and shut the gate on the other. Then knock it out, sail to safety and finish your long tame.
If you want to get a lot of pelt, make your base by the hidden lake on the Island, you can look up coords. Beavers spawn there, then harvest their bodies with pick. Don't worry about overhunting they just spawn again. And free dams.
I play on singleplayer, make sure the one you wanna tame is alone and doesn't have it's buddies arouns, bola it and knock it out usinf tranq arrows, took around 100 mejorberries and 20 narcotics. Time? Around 20-30 minutes.
The method I used to tame was picking up the beaver with my crystal wyvern and dropping it in a taming pen. Pretty easy knockouts but their torpor drop rate is INSANITY and you can't leave then for more than 3 minutes. Feed them either mejoberries or crops (kibble is really expensive but if you want to do that, go for it).