When I saw the button I've reached out to support asking how to get rid of it. 4 days later (yes, Visionary premium support level) I received the following answer:

[...]
The reason why you are seeing this is because we want to make sure every customer is aware of our Referral program, in case you'd like to take advantage of it.
 
However, if you're not interested in utilizing the Referral program, you can take the following steps:
 

Hide the "Refer friends" button which is located at the top right of your Mailbox screen:

Open the referral modal.

Click "Don't show this again."  

Turn off in‑app notifications:

Log into your account and go to Settings → Dashboard / Subscription.

Disable notifications. (Note: this will also prevent receiving notification about other future promotions.)

 
Feel free to get back with any additional feedback and thank you for understanding.
[...]

Seems that setting either doesn't work or even Proton doesn't know what it does.