Can I partially upgrade from Shimano Sora to Ultegra 6800? I have a Shimano 9 speed groupset and I want to upgrade to something that's better on the climbs without shelling out and buying a whole new groupset which would be more money than my actual bike.

Would it be possible to just buy a new 11 speed chain + 11 speed rear derailleur + 11 speed cassette + 11 speed shifters (all ultegra 6800)?

I.e. I want to avoid buying front derailleur + hub + crankset.  Would this work and be worth it?

Thanks!
Stay with 9 speed. the beauty of Shimano 9 speed is that the 9 speed road levers are compatible with mountain derailleurs and vice versa. You can get a 9 speed deore derailleur and a 9 speed 12-36 cassette and Bam! way more gear range. That's literally all you have to buy (and maybe a cable if your old one dies on removal) edit: you will likely need a new chain as well as your current one will probably be too short (but hey 9 speed chains are way cheaper than 11 speed chains!)

If you actually want to go to 11 speed, as stated above you'll need at least chainrings, chain, cassette, shifters, and front and rear derailleurs. It won't necessarily buy you more gear range though, honestly. I believe the ultegra 11 speed derailleurs can still only go up to 32 teeth on the cassette maximum. edit: as @nathan Knutson pointed out, 11 speed road for Shimano uses a wider freehub body (mountain does not, but is incompatible with drop bar shifters/road derailleurs) so you could not simple put an 11speed cassette on your current wheel