Persistent Audio Player
The only way to do that is to actually NOT refresh the page each time but rather use ajax or something similar to refresh only certain areas of the page while leaving the player area intact.

An example of this is pitchfork.com.  It has a player in the upper right gutter.  When you click on a link, what happens is the main content area of the page is reloaded but the page stays the same, so the player keeps on playing.

But, if you hit the reload button on your browser, it stops and starts all over again because the page itself was reloaded.

Of course, you could always put the player in a pop-up/under window and then you wouldn't have to worry about it as much.