If oil catches fire in the pan, the worst thing to reach for is the thing everyone reaches for.
Water is denser than oil, so it sinks straight through to the bottom of the pan, where the oil is well past 300°C. At that temperature water flashes to steam instantly, expanding around 1,700 times. The expanding steam blasts the burning oil upward in a cloud, and that cloud is what sets the kitchen alight. The pan was never the danger.
What works is dull. Kill the heat. Take away the oxygen. Slide a lid across the pan, or drop a fire blanket over it.
What doesn't: adding water, lifting the pan, or carrying it outside — a pan in motion sheds burning oil across every room it passes through.
Cover it, and leave it alone until it's cold.