What's the technical limit to how small RFID-like system antenna can be?
RFID tags can be made Damn Small™. Among the smaller ones are subdermal tags used for instance in animal identification:



IIRC in Belgium every horse must have such a subdermal RFID tag.
Other small tags (smaller than a credit card, anyway) are key fob sized tags  

  

and, about the same size, RFID tags that look like a wrist watch, I often see those in, again Belgian, cafés and restaurants to identify the waiter at the cash register computer.

RFID most often uses 125kHz or 13.5MHz carriers. If you go to 2.4GHz you can further miniaturize the tag. The following, 0.4mm x 0.4mm in size, is the smallest I could find:



So RFID tags can be made very small indeed. I guess there are much less miniaturization possibilities for the transmitter if it wants to extend the near field to a workable distance. The tag has to be in the near field to be detectable.