In The Wonderful Wizard of Oz, the Land of Oz (meaning, in this case, the four countries and the Emerald City) really doesn't seem too different from anywhere: animals can talk, and some of them are very strange, and people can do magic. But later on in the series, we discover that people cannot die.
Animals, however, are not so lucky. The Wicked Witch of the West's wolves, crows, and bees are all dead, or so we are told, as is a giant spider the Cowardly Lion defeated, and also a wildcat. We are also led to believe the Kalidahs that pursued our heroes also perished. In The Road to Oz, we are told a pet bear choked to death and was made into a rug before being brought back to life with a Powder of Life. (With a notably different formula.)
So, if we assume Oz was always this way, it gets problematic. The Tin Woodman tells us his parents died, while somehow he managed to become completely metal by his body being replaced by metal and—seemingly—his "liveliness" and soul being moved to this new body. So much that he could even be taken apart and have replacement parts made and still be a live metal man.
Baum never gives us another such character with such a strange history, except for Captain Fyter, whose story is almost verbatim Nick Chopper's origin again.
Thompson played with it a few times. She offers the explanation that the Scarecrow's body is now the new home of the "liveliness" and soul of Emperor Chang Wang Woe, which feels odd to me, as the Scarecrow never feels like how she described the Emperor. Perhaps the Scarecrow got the Emperor's "liveliness," but not his soul.
Bill, a weathervane rooster in Grampa in Oz, came to life after being struck by a "live wire." But more interesting is Dr. Herby in The Giant Horse of Oz. Mombi tossed him into a bottling solution which dissolved him into liquid, seemingly, which would seemingly prevent him from living. However, when the bottle his liquid was poured into was broken, he reconstituted into a live form. (Not exactly his original form, though.)
Few Oz stories I've read have played with this extreme again. It's more or less maintained (as established by Baum) that a person could be thoroughly destroyed, which would be as good as dead. However, this extreme is hard to pinpoint: a person chopped into tiny pieces would still be alive. What if they were burned to ashes? Or... digested?
An Oz book I'm reading now has another example. It's outside the Famous Forty +, so it's bearing on canon is subjective. It's The Seven Blue Mountains of Oz: The Disenchanted Princess of Oz by Melody Grandy. In one episode in the book, a man named Faraq falls into a vat of glass ready to be blown. He is presumed dead, but when a glassblower uses this glass, the glass forms into a crystal replica of Faraq, who is also alive. Like the Tin Woodman, Captain Fyter, and Dr. Herby, his body has been replaced with a non-flesh substance.
So it appears to me that as far as death in Oz goes, it is impossible for people to die, especially since the people of Oz have been enjoying a selective aging process since Ozma began her rule it seems. If an animal dies by some means, then it is dead. (As happens to one of Billina's chicks in The Emerald City of Oz, apparently animals are also not immune to disease.) However, if a body is unable to support life, the "liveliness" and soul of the person may enter another substance (sometimes, that which destroyed its body), and in the cases we know of, it attempts to take human shape. This now raises the question, wouldn't that mean that the "liveliness" (there's got to be a better term) and souls of many people who had "died" in Oz before Ozma's reign are either in new forms or floating about there?
... That's kind of creepy.