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Every single push-button start car I've driven has the ability of "engine off, radio on". On the downside my current car (2012 Mini Cooper S - push-button start) will only play the radio for about 10 minutes (??) after the engine is turned off.

On my vehicle (Ford Explorer) the radio doesn't turn off until someone opens the door.

My Mazda will keep the radio on if you turn the car off while it's in gear or neutral. Then you can put the car in park, and the radio will stay on until you hit the ignition button again.

That sounds like a bug.

Edit: after reading some Mazda forums, it looks like you can just hold the shift button (if you have an automatic) while on park to make the radio stay on, no need for neutral or other gear. You might want to try that.