I just got a Thinkpad P52s brand new for £800, it has the number pad, full hd screen, ssd, 16GB ram, some low voltage variant of the i7, USB-C charging, webcam, 3 year next business day on site warranty. It was cheap as the P53s has just come out, however looking online the previous P series all seem to be outside your budget (and mostly more than I just paid for a new one!).
Everything worked out the box on the p52s using Ubuntu 18.04 with no issues at all, even firmware updates "just work" using fwupdmgr. The only problem with this was that I bought it to replace her old T500 as that was getting very long in the tooth, having got it and had a play I was considering getting one for me to replace my W530 but the price went up to £850, so I considered it a bit more and now the last few are £1000 :(
Adam