False.

“A beaver’s nose has a special inner flap that seals out water when the beaver is submerged. Beavers have a special passage from their nose to their throat which connects to the upper lungs. This lets the beaver chew and breath through the its nose at the same time. Adult beavers can hold their breath under water for 15 minutes.” -Wildliferesponse.com

Other supporting links:
The Humane Society of the United States – Beavers: Nature’s Engineers
gpnc.org – Beaver

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