Is anyone else having this problem when you proceed with payment nothing happens? The animated processing indicator keep running but nothing after that.

This is when I purchase a gift for someone.

Is you bank using 3D secure or similar security measure ?

I have issues with one of my credit card where the 3D secure window never show up resulting in either the transaction failing or a behavior similar to the one you describe.

Yep, I encountered this last night: http://www.gog.com/en/forum/general/payment_system_broken

I haven't raised a support request, but I'll do it now.