Usually when a lightsaber strikes a person, the part of the body where the lightsaber struck is cut off; but when Vader's lightsaber strikes Obi-Wan, he just vanishes.
Is this because of Vader's lightsaber, or is it something else?
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This is unfortunately never sufficiently explained within the movies, shows, or Disney Canon, but this is not entirely unique, and has nothing to do with Vader's Lightsaber.
Upon death, some Jedi are able to become Force Ghosts (which we see at the end of Return of the Jedi.
Notably, we see the bodies of both Obi-Wan and Yoda disappear upon death, and Yoda's death was a natural one.
This is explained in the (Disney canon) factbook "Star Wars: Absolutely Everything you Need to Know".
Q. How does Obi-Wan reappear after Darth Vader defeats him?
A. After being struck down by Darth Vader on the Death Star, Obi-Wan becomes one with the Force. He appears as a Force spirit and is able to communicate with Luke and Yoda.
We see the same disappearance (and later reappearance) in the final film of the original trilogy when Yoda dies