Straight from the horse's mouth

You can't fight during a long rest.

"A Long Rest is a period of extended Downtime, at least 8 hours long, during which a character sleeps or performs light activity: reading, talking, eating, or standing watch for no more than 2 hours. If the rest is interrupted by a period of strenuous activity—at least 1 hour of walking, Fighting, casting spells, or similar Adventuring activity—the Characters must begin the rest again to gain any benefit from it."

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