Droids may not pull people's arms off when they lose, but Jedi sure do love cutting people's arms (and occasionally other limbs) off.
I don't think there are any examples in The Phantom Menace, but Attack of the Clones features Anakin's loss of hand (and I seem to recall a couple people and critters getting bits chopped off in the big arena fight). Then in Revenge of the Sith, Count Dooku loses both hands, Mace Windu has his lightsaber hand chopped off in the fight with Sidious, and Obi-Wan of course removes Anakin's remaining limbs before leaving him to die in miserable pain.
Then there's the random guy hassling Luke in the cantina scene, the Wampa on Hoth, Luke himself on Bespin, and Vader in the big fight in Jedi.