BEIRUT, Lebanon — Saudi Arabia and several other Arab countries that recently cut off diplomatic ties with Qatar issued a harsh list of demands on Friday, insisting that the wealthy but tiny Persian Gulf nation shut down the news network Al Jazeera, abandon ties with Islamist organizations and provide detailed information about its funding for political dissidents.

The demands, which were presented to Qatar through mediators from Kuwait, risked pulling other powers deeper into the rift by calling on the country to shut down a Turkish military base and to downgrade its ties with Iran — a difficult task given that Iran and Qatar share a large gas field that provides much of Qatar’s wealth.

The demands signaled an escalation in the deepest political crisis among Arab Gulf countries in years, after nations including Bahrain, Egypt, Saudi Arabia and the United Arab Emirates cut economic, diplomatic and travel ties with Qatar this month, accusing it of supporting terrorism.

Qatari officials did not immediately respond on Friday, but they have denied supporting extremists and said that they would neither negotiate while under a blockade nor submit to demands that undermined the country’s sovereignty.

All of the nations involved are allies of the United States, and most — including Qatar — host large American military bases. But analysts have accused the Trump administration of sending mixed signals, exacerbating the rift.

After the Arab nations announced that they were cutting ties with Qatar, Mr. Trump tweeted his support and even suggested he had been responsible for the move.

But that did not stop his administration from signing a previously approved deal for Qatar to buy $12 billion of American F-15 fighter jets.

As the crisis has dragged on, American diplomats have complained privately that the Arab nations were taking too long to present their demands, and Secretary of State Rex W. Tillerson said that they must be “reasonable and actionable.”

Qatar has historically played a maverick role in the Gulf Cooperation Council, a regional group that also includes Bahrain, Kuwait, Oman, Saudi Arabia and the United Arab Emirates. It maintains ties with a range of Islamist groups throughout the region, relationships that other countries have found useful when negotiating hostage releases but have complained about when those groups challenge their rule.

Qatar has also opened its doors to members of the Muslim Brotherhood, which Egypt, Saudi Arabia and the United Arab Emirates consider a terrorist organization; to members of Hamas, the Palestinian militant group; and to the Afghan Taliban. It has also financed Al Jazeera, one of the Arab world’s most watched satellite news channels, which is often critical of Qatar’s rivals.

Those stances have rankled others in the region, like Saudi Arabia and the United Arab Emirates, who see political Islam as a threat to their monarchies. And Qatar’s support for the Arab Spring uprisings and for the Muslim Brotherhood have angered countries including Egypt.

Other nations, like Turkey, have stood up for Qatar.

The Turkish defense minister, Fikri Isik, rejected the demand that Qatar close the Turkish military base and suggested that Turkey would enhance its presence there as a show of support.

“Strengthening the Turkish base would be a positive step in terms of the Gulf’s security,” he said, according to Reuters. “Re-evaluating the base agreement with Qatar is not on our agenda.”

Analysts said that many of the other demands would be hard for Qatar to meet, especially in the 10-day period the four nations gave Qatar to comply.

“I am very concerned that there is absolutely no way to de-escalate this,” said Michael Stephens, a research fellow at the Royal United Services Institute, a think tank in London. “I don’t see how a deal can be made at the current time unless something dramatic changes.”

Downgrading ties with Iran could complicate operations at the large gas field the two countries share, and closing the Turkish base would violate long-term diplomatic agreements, Mr. Stephens said. And Qatar is unlikely to give up Al Jazeera, a channel that has given it an outsize voice in regional politics.

More broadly, the list of demands called for undefined “compensation” and for Qatar to conform politically and economically with the other nations of the Gulf Cooperation Council.

If Qatar complies, the four countries said, it will be audited once a month for the first year and quarterly during the second year. It would then be monitored for compliance for the next decade.

Mr. Stephens, the analyst, said it was highly unlikely that Qatar would comply. “That would be a violation of sovereignty that the Qataris will never be able to accept,” he said.