Yes, the Master Sword does break in The Legend of Zelda: Tears of the Kingdom. This happens at the very beginning of the game. While Zelda and Link are exploring underground, they come across a mummy with powerful magic who breaks the Master Sword.
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How Does the Master Sword Break?
As the evil mummy wakes up, it attacks Zelda and Link with reddish-black Gloom magic. Link stands in front of Zelda and blocks the attack with the Master Sword. However, the Gloom magic is too powerful and it shatters the Master Sword—as well as severely damaging Link’s right arm.
After the cutscene, Link wakes up to find that he has a new arm and that the Master Sword has been decayed by the Gloom magic. Despite the legendary blade now being basically useless, Link hangs onto it for now.
What Happens to the Master Sword?
Link uses the Master Sword to make his way out of the ruins where he just woke up. When he does, he finds himself on the Great Sky Island. Then, Link meets Rauru and learns that he needs to visit three Shrines on the flying island before he can get into the Temple of Time and leave.
After doing so, right before Link leaves the Great Sky Island, a cutscene plays. Here, Link touches a ball of light and holds up the broken Master Sword. The warm, golden light envelops the Master Sword and it disappears.
When it reappears, it is in front of Zelda—who is nowhere near Link at this point. Zelda holds the broken blade close and closes her eyes before the cutscene ends.