OK, but that could be a fault with the battery/ external switch in the power connector, a fault in the power switching section of the main chip, a wire off inside, a PCB crack, the power button pads .....
Your biggest problem may be that in our throw-away, wasteful, cant-be-bothered-with-repairs society most retailers don't have anyone in-house who can diagnose faults let alone repair them, so they'd send your radio off to their service centre to be mended, with no option to say no if it's too expensive, and a charge for looking even if it's irreparable.