So, I’ve found a solution by consulting the support.
This is a WARNING for all surfshark users: When IPv6 is enabled, the VPN (which will connect via IPv4 as today) will sucessfully connect, but your traffic will run through the IPv6 from your provider (unprotected, not routed over the VPN server) nevertheless…
The solution is too either shoot birds with cannons and disable IPv6 altogether in /etc/default/grub by adding ipv6.disable to GRUB_CMDLINE_LINUX_DEFAULT , for example:
GRUB_CMDLINE_LINUX_DEFAULT="ipv6.disable=1 quiet splash"
Do not forget to invoke $ sudo update-grub afterwards.
Another (better way imho) would be to disable IPV6 in System Settings for the specific connection. Be sure to reconnect once, after disabling it.
Thanks to everyone for your input!