Cut Out the Old Faucet
Even with a basin wrench, it can be nearly impossible to break loose corroded nuts holding older faucets to the sink. If you don't care about wrecking the faucet, cut off the nuts instead. You can use either a rotary tool (Dremel is one brand) with a metal-cutting disc or an oscillating tool with a metal-cutting blade. Cut through one side of the nut. Then use a screwdriver to pry the nut away from the faucet body. You can also cut off other stubborn parts, like the pop-up drain assembly on a bathroom sink.