Hi @Stéphane, Yes, you can, but you'll have to do it yourself from your profile.
How does it work? Click on your profile, then edit profile < notifications and deselect the notifications you no longer wish to receive.
Hope this helps,
Cheers, 

Hi,

Thank you for your answer but it seems that I haven’t been clear enough.

>Yes, you can

I already know that I can configure the notifications. But how do I do specifically what I wrote: “keep notifications about private message but disable those about getting a new badge”
 

AFAIK messages about new badges are “Private messages” like any other.

 

Hi @Stéphane,
Unfortunately, it's not possible to turn off just the notifications about new badges. 
The only thing I can suggest is that you unsubscribe from all the emails sent by the Community Team via "Campaigns".
But I can see that you've already done this. If that doesn't work, there's no other solution.
Cheers,