Despite what car salesmen claim, yes, you can still negotiate, even if they claim the screen price is fixed and there's no haggling. We've haggled for every one of the 14 cars we've bought over the past 40 years of driving, including brand new, ex-demos, and used of various ages/mileages. Never paid screen/list price and never will. The last one we bought was a small brand new car around a year ago - list price was around £17k we got it for £14k - just 3 miles on the clock. The previous new car we bought was back in 2015 where we got a £20k list price car for £15k. In both cases, the car was sat in the dealership's lot and both were coming up to the time limit where they had to either pay for it or transfer to another dealership under the dealership financing loan scheme, so they were desperate to shift them otherwise they'd have to pay, and therefore "happy" to sell at cost price just to get rid - after all, they get registration bonuses so even if selling at cost, they still get money for registering it and it doesn't harm their cash flow. The car we bought between them was an ex-demo, just five months old, which we got for two grand under screen price despite the salesman being assertive that screen price was non negotiable and that it was on the internet at that price and "someone else" would pay full screen price within a couple of days because it was such a bargain! Funny how he changed his tune when he thought we were willing to walk away!
The key really is to make the salesman know and believe you're keen and in a position to buy, and more importantly, show him you're willing and happy to walk away. Just make sure they have your phone number. 9 times out of 10, you'll get a phone call the next day with them saying they've argued your case to the sales manager and he'll do an "exceptional" deal that the salesman has never seen before! It's all part of the game of brinkmanship! You just have to be realistic. In both the new cars I mentioned above, they weren't the cars we wanted, i.e. wrong colour, wrong spec, etc., but when you're getting that kind of discount, you have to make compromises. Had we been picky and wanted the exact colour/spec we went in looking for, then we wouldn't have got the deals as they'd have had to order from factory or found one to transfer in from another dealership, whereas they're keener to move what's in their lot or showroom.
Works at all levels of age/mileage, but haggling in used car garages, especially independents is a lot easier. Always helps if you know your facts, i.e. find similar cars in Auto trader or online to take to the used car garage to "justify" why you're offering a few grand under the screen price, by showing them a screen print of another garage with an almost identical car for a few grand less. Know your stuff and again, be prepared to walk away if they're not going to haggle to a reasonable compromise amount.