By Art Patnaude 

Even while international investors shun Greek stocks and bonds, there are a smattering of Greek assets still drawing interest: vacation properties on the island of Mykonos.

At this point, it's only interest. The escalation of the Greek debt crisis in the last week has drawn a rush of bargain hunters to the sunny playground of the global elite and A-list celebrities, but sales have largely dried up amid the increased uncertainty about the country's future, local estate agents say.

Greece and its creditors are scrambling for a solution to its five-year debt crisis after the Greek public voted on Sunday to reject tough new bailout terms. Given the political standoff, "people are mainly surveying the market [in Mykonos] rather than discussing specific properties," said Dimitris Manoussakis, head of Savills in Greece.

"There are limited transactions at this point. Nobody knows what is going to happen next, so there is very little confidence," Mr. Manoussakis said. Savills completed four high-end property transactions on Mykonos in 2015 before the recent upheaval ignited.

Still, home prices on Greek islands remain more resilient than everywhere else in Greece, where prices have fallen for six consecutive years and show no signs of letting up.

In Greece, estate agents estimate house prices have been cut at least by half since the 2007 peak. Official data from the Bank of Greece show a 40% decline since 2007, but given the very small number of transactions in the market, the figures "may understate the degree of actual value declines," according to Andrew Currie, managing director at Fitch Ratings.

Fitch doesn't break out numbers by island, but in Mykonos, local agents say prices are still down about 30%, having stabilized, and by some accounts picked up slightly, during a tentative recovery in 2014.

"Buyers haven't completely backed off yet" in Mykonos, said Roi Deldimou, manager of Beauchamp Estates office on the island. Ms. Deldimou is currently working to close two sales in the low end of the Mykonos market--deals valued at a bit more than EUR500,000 ($555,750)--for an Italian and a Greek American. The political upheaval has thrown a wrench into the works, but "they're still trying to figure it out," she said.

Luxury housing developer Euroterra Capital--based in London and founded by Greeks--is planning to open an office in Mykonos in September, said Pantazis Therianos, managing partner at the firm. Strong demand from tourists for rental homes on the island "supports prices in this market, " Mr. Therianos said. Euroterra plans to sell its developments to its clients in London and Asia.

Mykonos--immortalized by visits from Jackie Onassis in the 1960s, and just 35 minutes from Athens by plane--attracts the same ilk of foreign buyers as other European vacation spots, from the French Riviera to Tuscany and the Swiss Alps. Many of these markets have also suffered significant drops in prices since the financial crisis.

While more stable than elsewhere in the country, prices aren't shooting skyward in Mykonos. As transactions dry up amid the swift re-escalation of the debt crisis over the past month, the tentative recovery on the island could be extinguished, agents warned.

Bargain hunters have swooped in looking for cheap deals, but most sellers, unwilling to relinquish their stunning Mediterranean vistas for a song, are standing firm on asking prices.

The bargains are also relative. Homes listed in Mykonos are typically more than EUR1 million, said Robert Guy Danon, managing director at Danon & Co., an associate of Chestertons International.

Part of the reason there are so few sales at the moment is because "people know they can always rent [their houses] out," Mr. Danon said.

Ms. Deldimou at Beauchamp said one client who couldn't sell his villa, listed at EUR4 million, decided to rent it out for EUR6,000 a week. Even with money tight, "he just rented it instead of selling it," she said. The home remains on the market at the original asking price.

Buying property at this point in Greece isn't for the fainthearted. For one, there is the risk the property at some point might be valued in drachmas, Greece's currency before it joined the eurozone, if the country exits the single currency.

Greek property taxes and transaction costs also can hit buyers. For instance, on a EUR250,000 property, "the total amount at the end of the story when you added stamp duty, taxes, and everything else goes up to EUR400,000," said Ana Vukovic, managing director of Greece and Serbia for U.S.-based real-estate firm Colliers International.

"When you see an attractive property price, you need to be aware of how much you need to pay to the Greek state," Ms. Vukovic said.

Write to Art Patnaude at art.patnaude@wsj.com