Here is a description of how one guy extracted the electronics from a smart card and then embedded it into a ring wearable on a finger.
The card in question is more or less similar to the card in another question, the minor difference is that the card in the current question uses a thin (presumably copper) wire as antenna and the card in the linked to question uses wide thin flat traces. The wires run along the card perimeter. The card is credit card size, so the antenna is more or less an ellipsis.
So that guy uses solvent to soften the card plastic, strips the card and extracts the electronics. Then he rewinds the antenna wire so that the antenna coil can be embedded into the ring.
The resulting coil has several times smaller diameter yet the electronics work.
How does it happen that rewinding the antenna into a smaller loop does not break the card operation?