On some vehicles the power windows and radio stay powered until a door is opened, key doesn't matter.
This is tied in with the door latch switch.
To see if this might be the problem just check and see if the "Door ajar" light comes on when Drivers door is opened, if not then the switch in the Drivers door is not working.
Then test if Radio goes off if you open the passenger door, if so then you have the "leave power on" option and drivers door switch isn't working.
Google: Ranger Door ajar light
Switch sticking open is the most common issue.