Basecamp datafile reportedly corrupted, and no way to restore neither from time machine backup nor from an explicit backup made from the app.

First thing I'd try is uninstalling and reinstall BaseCamp as it's unlikely all of your databases are corrupt so BC may be reporting it incorrectly. If that doesn't work change the language to English and see if that helps, in fact you might want to try that first.

Well, I agree it's unlikely. 

I'll try to change the system language to english, but in the meantime, I tried to run the software from the command line, and got the following note (repeated many times) and error (once, after loading the backup, before the application restart) : 

2021-11-14 15:40:52.766 Garmin BaseCamp[12492:355667] CoreText note: Client requested name ".SFNS-Regular", it will get Times-Roman rather than the intended font. All system UI font access should be through proper APIs such as CTFontCreateUIFontForLanguage() or +[NSFont systemFontOfSize:].
2021-11-14 15:40:53.019 Garmin BaseCamp[12492:356805] ERROR: No replacement for {1} (/Users/buildy/Desktop/BaseCamp/mac-mapinstall-4.3.5-source/mac/libs/Quadrinity/QDBackupRestore.mm: 308)

A new attempt with the system set to use english as primary language results in the same error, with a few refinements : 

2021-11-14 15:48:25.794 Garmin BaseCamp[12715:364964] CoreText note: Client requested name ".SFNS-Regular", it will get Times-Roman rather than the intended font. All system UI font access should be through proper APIs such as CTFontCreateUIFontForLanguage() or +[NSFont systemFontOfSize:].
2021-11-14 15:48:26.605 Garmin BaseCamp[12715:366119] ERROR: No replacement for {1} (/Users/buildy/Desktop/BaseCamp/mac-mapinstall-4.3.5-source/mac/libs/Quadrinity/QDBackupRestore.mm: 308)

after restart : 

2021-11-14 15:49:45.366 Garmin BaseCamp[12763:367238] CoreText note: Client requested name ".SFNS-Regular", it will get Times-Roman rather than the intended font. All system UI font access should be through proper APIs such as CTFontCreateUIFontForLanguage() or +[NSFont systemFontOfSize:].
2021-11-14 15:49:45.366 Garmin BaseCamp[12763:367238] CoreText note: Set a breakpoint on CTFontLogSystemFontNameRequest to debug.
Segmentation fault: 11

I'll try a full reinstall... 

Just in case you haven't tried this, how far back did you go in Time Machine? It may be worth trying to go farther back to find an uncorrupted file.

Full reinstall with the latest version did not help either... I will try and restore a former version to check. 

The last (important) modifications I made ere on 16/10... this is also the day I made a backup (just in case). Too bad it does not help... :-/  

Ok, I managed to manualle extract the content of the backup, which happen to be a simple zip file. 

When I try to import AllData.gdb (either version 4.7 or 4.8, and even the .bak) I get : 

AllData.gdb corrupt and could not be imported.

Now, is there a way to diagnose and repair this gdb file ? 

I would go back farther in Time Machine, as danham_ has suggested, and use it to replace that entire '4.8' folder.

If replacements from TM backups as far back as you can find a '4.8' folder still show as corrupt, perhaps rename your '4.8' folder as '4.8x' and rename your '4.7' folder as '4.8' and restart BaseCamp. This method will probably lose some recent data, but could get you back up and running.

I will try this, but unfortunately, the data I need to recover is from octobre 16th, so any further back in time will be of no use to me. 

Understood. But it would be useful for diagnostic purposes. IOW, if an old "useless" Time Machine backup loads and works properly, you now know that it is indeed the data file that is corrupt and you are basically hosed. BUT, if it does not work properly, that may point to other types of corruption that BC is incorrectly flagging as data file problems. Like maybe preferences or other system-level stuff. To test this, try a fresh install of BC  in a new user account on your Mac and then test with an old data file and if that now works, with your latest file.