It turned out not to be a problem with Docker at all. We had a request filter that added the Secure-flag to cookies that are sent with the response, but only to requests that comes from remote machines. The Docker container is in this case seen as a remote machine.
When testing on my local IIS I am doing this over HTTP, which means that secure (HTTPS) cookies are not sent.
Unsure if this is Docker, Hyper-V or IIS-specific.
I am using Docker for Windows (beta, with Hyper-V) on my PC. On the same PC, I am using IIS to serve a website.
When visiting the IIS website from a Docker container, the Chrome instance in the container does not receive any HttpOnly-cookies (I have used VNC to check). It does get normal cookies though, and normal traffic works fine.
If I visit the website in Windows from the same PC, I receive all cookies.
Both my Windows and the container has the domain for the website in their HOSTS-file (although different IPs, both of which has a binding to the domain in IIS).
I think this worked when using VirtualBox with Docker - but I cannot be 100% sure.