Gold Mouse

All the gold critters are long odds, but this one is worse than the rest because of where it lives. Mice are underground, which means every attempt happens somewhere that is actively trying to kill you, and the usual trick does not apply either: town NPCs do nothing for mouse spawn rates, so there is no way to stack the deck.

The workable approach is to stop hunting and let it come to you. Keep a Lifeform Analyzer running while you mine and it will flag one before you have walked into the room. Nothing else in the cave can hurt it, so the only thing that will cost you the catch is your own weapon or a careless swing of the pickaxe.