There once was an AUR package but it no longer works. 
The free implementation of BankID FriBID also no longer works with the current version of BankID.
Running BankID in a virtual Windows machine
Install VirtualBox, download a Windows ISO, setup the machine and install BankID. If you want to use a card-reader you have to add your user to the group
vboxusers to be able to access the usb-connected card-reader. You can then add a usb-filter in the Virtualbox settings for the machine, so that the machine recognizes the card-reader automatically.