How to stop the spinning rainbow wheel on an Mac without losing data?

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

EtreCheckPro is an excellent tool for identifying both hardware and software issues.

Download and run the free version of EtreCheckPro, from > https://etrecheck.com/en/index.html

Then post back here with your Report, as per > How to use the Add Text Feature When Post… - Apple Community

Does your keyboard still work?

If yes, did you try to close the stuck app by pressing Command + Q

In case your app is completely unresponsive, you will have to force quit the app by pressing Option (or Alt), Command, and Esc.

This will open the Force Quit window and you can choose to close the unresponsive app, but unsaved data will be lost.

If the Mac is not responding to any keyboard inputs, you will have to shut it down manually by pressing the Power button until the display and the Mac shuts down. Unsaved data will be lost as well.

More details can be found in this related article:

How to force an app to quit on Mac - Apple Support