Yes - I got this working. But it turned out that the problem was most likely caused by some changes to the network configuration, so I am not sure my ‘solution’ will work for you.
I discovered that the users from domain B was using a different url when accessing the QV AP; something like ‘http://QVServer.DomainA.internal/qlikview/index.htm’. And it was only when using this, the users were prompted for user name/password.
The regular users were using ‘http://QVServer/qlikview/index.htm’. When I had the users from domain B use the regular url, they were not prompted.
I suspect that there is a good explanation for this, but unfortunately I never got to the bottom of why the access point url was constructed like this in the first place and if some later network change caused the problems.
Thanks for all your help everybody – most appreciated.