OK - I got it. After digging into this thread: After today's update, Chrome-based browsers (Chrome, Opera, Vivaldi) will not open web pages I tried stopping systemd-resolved and voilà: all braves are brave again 
. There exists a nsswitch.conf.pacnew but that does not solve the issue. I cannot say where the issue resides but stopping systemd-resolved fixes the problem. For me only the mentioned brave browsers are affected. All others worked fine.
Edit:
To be more precise:
I changed one line in /etc/nsswitch.conf and everything works (even with systemd-resolved running):
From
hosts: files mymachines myhostname resolve [!UNAVAIL=return] dns
to
hosts:files mymachines myhostname dns resolve [!UNAVAIL=return]