This is a pretty big change.
Generally, the idea of master pages is that you can have many pages (aspx) in your site that all have the same "chrome" around them that is provided by the master page layout. The individual pages just change the contents of the ContentTemplate. You can't make the master page keep anything "static" in the browser unless you completely change your design.
What you would need to do is change the architecture such that instead of many pages (aspx files) you have many controls (ascx files) that act as the pages. You then have a single main page (in this case you don't even need a master page, because you will only load one aspx file). The main page has an update panel in it and as you changes "pages" (controls) you dynamically load the appropriate control into the page body. This is the only way I can imagine meeting your goal. You have to load the master HTML set with the media player in it and then never do a full post back or page reload so you have to do everything with AJAX in an update panel.