Don't judge me but I used a butane lighter to heat and unscrew one. Yes it was probably paraphernalia but the tenant left it where I could see it and it worked. YMMV

Arokroil or pb blaster. Did you try tongue and groove pliers? Use rag or something to protect chrome, but some damage might still happen.

Get a flat-bladed screwdriver, rest the tip in one of the notches, and tap the handle gently with a hammer. Alternate occasionally between notches to reduce cumulative damage to he faucet stem. The shock should loosen the nut.

Find the replacement nut online. You'll feel better if and when you destroy it breaking it loose. I just did my Moen shower cartridge (with a "lifetime warranty") which was so bad I had to cut the decorative shower handle off to break the screw loose....point is the new part was sitting on the counter so I felt better about shredding the existing part.

Yeah, those buggers can be feisty. One thing to try is get a hold of it with channel locks, put a bit of pressure in the loosening direction, then give the handles of the channels a little tap with something; an actual hammer might be overkill, but anything to send a shock through the threads. Crescent wrench, screwdriver handle, lots of things can do the trick. We're basically breaking any dried schmoo in the threads.

Another piece to keep in mind is the part you're trying to turn is actually a bit flexible; if you squeeze too tight, you can pinch the outer threads against the inner threads, stopping it from turning. Adjusting the position of the pliers can make a big difference.

A comforting thing to know here is the faucet probably has several parts going through the sink; this means you can really put some sauce on to get it to break free without the faucet turning on you. Keep after it; it'll let go!

Edit words are hard

i always hate messing with those, usually i just say screw it and just buy a new entire faucet, that way i just have to deal with the undersink screws and hoses, which i know can be annoying too, but with a decent basin wrench i can be done in 20 min, instead of messing with a rusted collar for hours or days.

Vinegar will work like a charm if you have hard water (white calcium deposits) but it looks like some rust. Like someone else said, use some channel locks with light pressure and use a rag on the teeth to prevent it from scratching the chrome. Clean it really well afterwards.

PB Blaster. Give it time to work. Use channel locks with a strip of cloth to prevent scratching.