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Yes you can. Chances are real strong that the metal 'hook' on the end of the nylon cord broke. The nylon cord wraps around a drum inside (the motor you hear whirring rotates the 'drum'). And as the drum unwinds, the semi-rigid nylon cord unspools and pushes the antenna up. And there's a metal 'hook' attached to the end of the cord which hooks into the drum (so the cord has something to attach to). When the hook breaks off, the nylon cord is not really attached to the drum anymore and will just spin freely while the drum turns (what you're hearing). You can drill out the rivets on the bottom antenna body and disassemble and reassemble with small stove bolts. There's a real good chance you'll be taking it apart again, so bolts work better than rivets. It's one of those jobs you wish you had a 3rd hand.
But you should take a look first before you buy any parts, in case the problem lies elsewhere. Other things can go wrong too.