busy as a beaver/bee
Extremely industrious. The proverbial comparison to bees dates from Chaucer’s time. The one to beavers is newer, going back only to the seventeenth century; it also is put as works like a beaver and eager beaver. Among more recent proverbial comparisons for being busy, which liken it more to nervous overactivity than ambitiousness, is busy as a one-armed paperhanger, an Americanism dating from about 1910.