One who or that which gnaws; a creature that bites or scrapes with its teeth.
From gnaw plus -er agent suffix. The -er suffix creates nouns for things that perform the action of the verb, so a gnawer is literally 'one that gnaws.'
Rodents are perhaps the most famous gnawers, and their front teeth never stop growing—they literally have to gnaw constantly just to keep them from becoming too long to eat with!
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