Critics say the Nordic island country should also have closed all of its schools and stopped tourism. But the government says it believes that new cases have peaked.

With its small population and isolated location, Iceland has earned praise and headlines for its plan to test as many people as possible for exposure to the new coronavirus. Why, some wondered, couldn’t other countries be like Iceland?

But critics inside the country have called this rosy picture misleading.

They say the tiny Nordic island country of 360,000 people has not done enough to suppress new cases of Covid-19, the disease caused by the virus. Primary schools and day-care centers remain open, as do some restaurants with limited seating. Tourists are still allowed to arrive and travel without quarantine. The authorities at first limited gatherings to 100 people, then changed that to 20, long after other countries were imposing greater social isolation.

Iceland’s goal of testing everyone faces the same logistical hurdles that all countries face, the critics point out. It does not have enough medical personnel, supplies or hours in the day to test hundreds of thousands of people in a few weeks or months. They have warned of false optimism that will ultimately lead to more infections and death.

But Kjartan Hreinn Njalsson, the assistant to Iceland’s director of health, touted what he called a landmark moment on Wednesday: “More people are now getting better than getting infected.” In fact, he said, government officials believe cases may have peaked. “We are not over the hill, but close to it.”

Mr. Njalsson also said the country is well stocked with testing swabs and other necessary materials. “We know that we will eventually need more viral swabs,” he added, “and that every country in the world is trying to secure stocks.”

But so far Iceland has been steadily testing people with and without symptoms, with the help of deCODE genetics, a company that has long operated in the country and is a subsidiary of Amgen. The country has one of the highest proportions of tests performed by any country for the coronavirus, according to government officials. They have tracked the origins of different infections, traced contacts and even looked at the virus’s different strains.

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As of Wednesday, at least 30,000 samples had been tested, according to officials, and the country had at least 1,600 confirmed Covid-19 cases. Iceland confirmed its first case on Feb. 28. Six people have died.

To get tested at a hospital or clinic, individuals must meet certain criteria, such as having symptoms like a fever or being a member of a high-risk group. Health care workers, people who have traveled from high-risk areas, and those in quarantine are also able to get tested at a hospital or clinic.

Nearly everyone else can sign up to be tested by deCODE.

Dr. Kari Stefansson, the chief executive of deCODE, said the testing and sequencing that his company had provided indicates that, with a cohesive screening approach, “you can slow the spread of the virus, you can make it manageable for society.” The company has called for testing asymptomatic people as a way of finding undetected cases and stopping the spread of the disease.

Dagur B. Eggertsson, the mayor of Reykjavik, credited the country’s large-scale testing and daily briefings about the results with getting average citizens engaged with preparations. “We can’t use force,” the mayor said. “We have to use arguments and voice of reason to get people to work together.”

But a few dissenters have said extreme restrictions were exactly what Iceland needed to stop the outbreak entirely. They warned, for example, that keeping schools open could contribute to the virus spreading.