Is there a way to make paper reflective like a mirror?

First figure out how thick you want your reflective coating to be and obtain an aluminum ribbon of that volume. Attach the ribbon to electrodes inside a vacuum chamber. This must be a glass or plastic chamber, not metal. Place the paper near the wall of the chamber, approximately parallel to the ribbon. Attach a high voltage power supply, positive to the ribbon and negative to a plate on the outside of the chamber.

When the chamber has a high vacuum, turn on the high voltage and pass a large current through the ribbon. The aluminum will be vaporized and attracted toward the negative terminal, but trapped by the paper instead.

This is the method used to make vacuum tubes. If you have seen one you may have noticed the patch of reflective metal on one side of the tube. That is an aluminum flash with the remaining air molecules trapped in it. You may be able to find a custom tube manufacturer who can do this project for you.

Cop out method: glue aluminum foil to the paper.