For a permanent/persistent audio player I can think of these options:
- a completely Ajaxified theme, so the content is dynamicaly loaded
- put the audio player in a frame and the navigation/content in another one (yes, frames, that thing of the past)
- open the audio player in a pop-up
- make WordPress the CMS but pull the content in a Flash site (visitors see Flash, Google see WP)
Otherwise, it’s a completely unfriendly player: at each navigation link clicked the thing restarts playing…
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This is the simplest plugin I know and it’s easy to style:
http://wordpress.org/extend/plugins/haiku-minimalist-audio-player/
You can find simple and free flash players here:
http://ffiles.com
For feeding a Flash player with WP content, open a new question and drop me a line, as I have written many versions of this kind of interaction.