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I've got a '62 1/2 ton with the older style full size bed on it. Looks kinda strange, but it came that way from the factory. I am not sure how many of those were built like that, especially since it is not 4 wheel drive.
I believe that from '61 thru '66 the "old-style" ('57-'60) style separate box was standard equipment for the styleside version of the F-100 and F-250 4X4 and the F-350 2X4 all of which had 8-foot beds. The wheelbase of these trucks varies slightly from the standard F-100 longbed wheelbase. A 9-foot bed was also available on the F-350 and was offered with the same '57-'60 style styleside box.
Is that "rare duck" to which you refer the '61 F-100 from Argentina? That truck may have been manufactured in South America without adherence to Ford model packaging. I don't think any American-made F-100's (2X4) were separate cab/box style in '61.