Well, it used to depend on what chat service/client you were using and how you're chatting with the friend.

I don't use gmail for chatting, but knowing the way Google operates, I'd venture to guess that you probably cannot get the IP address.

For things like AIM (and I think this has changed since), the way to obtain the IP address of the person you're chatting with was to make a 'direct connection' which basically bypassed the AIM chat server and connected client-to-client. This would obviously open up a connection directly to the IP of the person you're chatting with.

So, the simple answer is: no.

As the connection is typically: you <-> chat server <-> friend
Unless you can direct connect: you <-> friend

Hope that helps. Maybe someone else will have some further insight.

BillV

If it works similar to how Y! messenger works (or used to) you could like they said attempt to make a direct connection. You could do this by inviting the user to your web cam, inviting the user to photo sharing, sending a file, etc. After doing that you could run a netstat command and it's typical you'd see their IP Address next to the port that the messenger service uses. For Y! My Guess Would Be 5050, so I'd run : (Doing it as if your on a windows box)
netstat -an |find "5050"
In hope of finding my victim IP Address. Good luck though, your best bet would be to do it like how ethicalhacker said it, send them to a site.