Morale and striking
Ships don't sink. They strike.
A crew's morale falls as it takes casualties. Losses wear it down for good, and a sudden heavy loss, such as a rake, shakes it further for a while; that shock eases in a lull between broadsides, slowly. A mast coming down is a heavy blow to morale too, though it kills few, and part of that blow never comes back; on its own it never makes an unhurt crew strike, but it can finish one already hurt. When morale breaks, the ship strikes. Historically, the losing ships in single-ship actions struck when about a third of their crews were killed or wounded, and sooner if they were raked.
A ship that strikes stops firing, furls her sails, hauls down her colours and shows a white flag, and drifts. She keeps her side, and nothing fires on her. Her label in the glass says Adrift and no longer shows her captain's name: she has lost her commander.
If your ship strikes, a notice says so. The battle goes on, and you can still look around and watch your side fight on. The pause menu (Esc) offers Fight again and Leave.