Did the Castaways use paper money? Yes.
In early episodes, we see Mr. Howell hiring various services from other castaways. We eventually learn he’s been writing checks on a mainland (and therefore inaccessible) bank. This works while the group consider their condition temporary, but the checks are quickly devalued and eliminated when the castaways begin to prepare for the possibility of an indefinite stay on the island.
In Episode 9, “The Big Gold Strike,” Gilligan and Mr. Howell find a gold mine on the island, which Howell convinces Gilligan to keep secret from the others. By the time everyone learns about the mine, Howell has already taken the lion’s share of the most easily accessible gold. He’d like to hoard it for himself, but the other castaways begin charging him for their goods and services. Soon everyone has a small fortune in gold, which they all try to smuggle aboard a tiny escape raft. Their collected wealth, of course, ends up at the bottom of the lagoon.
In later episodes, monetary exchange takes place in US paper currency.
Might such fiat dollar exchange occur in the real world?
The market can reclaim money from government…This is why the castaways value Thurston Howell’s paper dollars: because whatever absurd amount he may have brought with him for “a three-hour tour,” that amount is now fixed. Dollars are the most stable currency available on Gilligan’s Island, and the government has nothing to do with it. Or rather: the absence of government has everything to do with it. If people are allowed to pick their own preferred money, they will pick whatever holds its value most reliably.
Here is the full analysis, courtesy of the Mises Institute.
My take: Fiat money would not survive on Gilligan’s Island, at least assuming that no rescue is expected. Skipper and Gilligan will return to barter, based on some useful island commodity, probably a storable foodstuff. Perhaps they will continue to use dollars as a medium of reckoning, for mutual convenience. But why accept dollars as a medium of payment? Given that only seven people live on the island, there is market power and room for dollar prices to slide over time. (To see one potential problem, imagine the polar case of only two traders — would you mind losing all your dollars? — surely prices would then fall in response.) Dollars would offer little stability of real value, if only because you might start to doubt whether other people would continue to accept them.