Vietnam has said the computer-chipped passports violate its sovereignty, and has demanded Beijing withdraw the documents, which show the contested Paracel and Spratly Islands as Chinese territory.
"We do not stamp the new Chinese passports," said an official at Hanoi's Noi Bai Airport, the country's main international gateway.
"We issue them a separate visa."
A border guard in northern Lang Son province said they were also not stamping the new passports, but issuing separate visas to Chinese arrivals.
Even with the new passports, however, "Chinese citizens can still travel normally through the border gate," the guard added.
Beijing has long infuriated southern neighbours such as Vietnam with its claim to vast swathes of the South China Sea, with Chinese maps showing a "nine-dash line" that runs almost to the Philippine and Malaysian coasts.
Both the Philippines and India have also protested against the map in Beijing's new biometric passports, and Taiwan last week protested after China started issuing the new travel documents with maps featuring two of the island's most famous scenic spots as part of Chinese territory.
The United States plans to raise the controversial new passports with Beijing, saying the documents were not helpful.
"We do have concerns about this map, which is causing tension and anxiety between and among the states in the South China Sea," State Department spokeswoman Victoria Nuland said.
"We do intend to raise this with the Chinese," she said, adding the new passports were not "helpful to the environment we all seek to resolve these issues."
Ms Nuland said the United States would accept the passports as a valid travel document, as it was up to countries "to decide what their passports look like as long as they meet international standards."
But she argued the look of a document was different from "taking steps that antagonize countries that we want to see a negotiation happen with".
India has started stamping its own map onto visas issued to Chinese visitors as the map shows the disputed border areas of Arunachal Pradesh and Aksai Chin as part of Chinese territory.
Manila, which claims part of the Spratlys, sent Beijing a formal protest letter last week, calling the maps "an excessive declaration of maritime space in violation of international law".
The South China Sea is strategically significant, home to some of the world's most important shipping lanes and believed to be rich in resources.
Other claimants to parts of the South China Sea are Brunei, Malaysia and Taiwan.
Beijing has attempted to downplay the diplomatic fallout from the recently introduced passports, with a foreign ministry spokeswoman saying the maps were "not made to target any specific country".