first i would try is checking if r8168 is installed pacman -Qs rt8168 pacman -Qs r8168
if so remove it.
Second would be to check if device is working with LTS kernel: sudo pacman -S linux-lts linux-lts-headers
reboot will boot on the LTS kernel
You are right, there is no “t” in that name
pacman -Ss r8168
community/r8168 8.049.02-38 [Installiert]
A kernel module for Realtek 8168 network cards
community/r8168-lts 8.049.02-34
A kernel module for Realtek 8168 network cards for linux-lts