I tried listening to the radio on the BBC site but can't reproduce that fault. What audio player is being used to play audio files?
After Firefox appears to have shut down (but really hasn't), can you see the audio player listed in the Running Processes as well as Firefox? If so, maybe it's something to do with that not closing down that is affecting Firefox.
Are you using the latest version of Firefox (0.9.3)? If not, you could try uninstalling your present version and upgrading.
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