Sounds dead to me. U4 probably developed a short circuit. Some boards do have polyfuses which get hot in case of a short circuit and won't allow the board to turn on again until after they cool down. However, U4 is not a polyfuse and the fact that U4 gets very hot is a dead giveaway that U4 is dead.
Update from comments: apparently U4 is a switch which is part of the power supply circuit. In this case it is also possible that something downstream of U4 has a short circuit. That component is probably also getting hot. That could be the main CPU U8 in which case the board is useless and you should get a new one.
If you are lucky, other parts of the board may still work, including the actual STM32 chip U8. The busted U4 is stopping them from getting any power because it's a short circuit. You'll certainly have to remove U4 to get anything to work again. If U4 has something to do with the USB connection then you might not be able to use the board anyway. If U4 has something to do with the power supply then removing it could cause further damage. No way to predict that without seeing the schematic or at least seeing what kind of chip U4 is.
If you want to get actual work done it is probably more productive to get another board. It seems they are not very expensive. If you value your money more than your time, you may remove or replace U4 just to see what still works.
According to the manual, the top part of the PCB is intended to snap off (hence the cut line) and you can reconnect it with a few wires. So if the main part is working you could still power and program it using the top part from another board. If you value your money more than your time. Note that the ADC unit in U8 is likely to be dead regardless.
STM
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