There are only three ways to go here. PHP alone isn't going to do it.
1. Use "internal page" iframes for the whole site (bad idea).
2. Use a very simple javascript line to open a new small window that contains your player embedded into a simple html page.
3. Build a "one-page" site and call all your content with ajax or jquery. Hmmm.
4. (Did I say three?) I believe there's a way to do this with Flash (again, somewhat of a "one-page" scenero with even less SEO value - another bad idea).
I opted for #2. You can control the look of your player and most of the attributes of the new window (size, position, toolbars, scrollbars, etc.) with this method. When you focus on your site's window, the new window drops behind your site's window and the user can bring it back up (to use the player controls) by clicking on it's tab in the taskbar. The user can go anywhere on-site or off-site and the player will keep playing seemlessly until they decide to close the player window or pause/stop the player.