Fixed header?

Short answer, yes, almost anything is possible.

Long answer, how familiar are you with web development, php, and js/ajax as it would require a good knowledge and a bit of development to implement a custom solution to this. I’m not aware of any plugins, but I would have allowed you search for such.

The best and easiest way to do it would be to have a small popup media player, a number of radio stations provide their online streaming services this way as a not to confuse the user, and so that they don’t accidentally navigate away from the browser window.