NAIROBI, Kenya -- Terrified shoppers huddled in back hallways and prayed they would not be found by the Islamic extremist gunmen lobbing grenades and firing assault rifles inside Nairobi's top mall Saturday. When the coast was thought to be clear, crying mothers clutching small children and blood-splattered men sprinted out of the four-story mall.

At least 39 people were killed and more than 150 wounded in the assault, Kenya's president announced on national TV, while disclosing that his close family members were among the dead.

Foreigners were among the casualties. France's president said that two French women were killed, and there were reports of American citizens injured, but the U.S. State Department said it had no further details.

Early Sunday morning, 12 hours after the attack began, gunmen remained holed up inside the mall with an unknown number of hostages. President Uhuru Kenyatta called the security operation under way "delicate" and said a top priority was to safeguard hostages.

As the attack unfolded shortly after noon Saturday, the al Qaeda-linked gunmen asked the victims they had cornered if they were Muslim: If the answer was yes, several witnesses said, those people were free to go. The non-Muslims were not.

Somalia's Islamic extremist group al-Shabaab claimed responsibility and said the attack was retribution for Kenyan forces' 2011 push into Somalia. The rebels threatened more attacks.

Al-Shabaab said on its Twitter feed that Kenyan security officials were trying to open negotiations. "There will be no negotiations whatsoever," al-Shabaab tweeted.

As night fell in Kenya's capital, two contingents of army special forces troops moved inside the mall.

Police and military surrounded the huge shopping complex as helicopters buzzed overhead. An Associated Press reporter said he saw a wounded Kenyan soldier put into an ambulance at nightfall, an indication, perhaps, of a continuing shoot-out inside.

Witnesses said at least five gunmen -- including at least one woman -- first attacked an outdoor cafe at Nairobi's Westgate Mall, a shiny, new shopping centre that hosts Nike, Adidas and Bose stores. The mall's ownership is Israeli, and security experts have long said the structure made an attractive terrorist target.

The attack began shortly after noon with bursts of gunfire and grenades. Shoppers -- expatriates and rich Kenyans -- fled in any direction that might be safe: into back corners of stores, back service hallways and bank vaults. Over the next several hours, pockets of people poured out of the mall as undercover police moved in. Some of the wounded were moved out in shopping carts.

"We started by hearing gunshots downstairs and outside. Later we heard them come inside. We took cover. Then we saw two gunmen wearing black turbans. I saw them shoot," said Patrick Kuria, an employee at Artcaffe, the restaurant with shady outdoor seating.

Frank Mugungu, an off-duty army sergeant major, said he saw four male attackers and one female attacker. "One was Somali," he said, but the others were black, suggesting that they could have been Kenyan or another nationality.

Al-Shabaab, on its Twitter feed, said that it has many times warned Kenya's government that failure to remove its forces from Somalia "would have severe consequences." The group claimed that its gunmen had killed 100 people, but its assertions are often exaggerated.

"The attack at #WestgateMall is just a very tiny fraction of what Muslims in Somalia experience at the hands of Kenyan invaders," al-Shabaab said. Another tweet said: "For long we have waged war against the Kenyans in our land, now it's time to shift the battleground and take the war to their land #Westgate."

Al-Shabaab threatened in late 2011 to unleash a large-scale attack in Nairobi. Kenya has seen a regular spate of grenade attacks since then but never such a large terrorist assault.

Nairobi's mortuary superintendent, Sammy Nyongesa Jacob, said Africans, Asians and Caucasians were among the bodies brought to the mortuary.

The U.S. State Department condemned "this senseless act of violence that has resulted in death and injury for many innocent men, women, and children."

The U.S. embassy said it was in contact with local authorities and offered assistance. Some British security personnel assisted in the response.

The gunmen told hostages that non-Muslims would be targeted, said Elijah Kamau, who was at the mall at the time of the midday attack.

"The gunmen told Muslims to stand up and leave. They were safe, and non-Muslims would be targeted," he said.

Jay Patel, who sought cover on an upper floor in the mall when shooting began, said that when he looked out of a window onto the upper parking deck of the mall he saw the gunmen with a group of people. Patel said that as the attackers were talking, some of the people stood up and left and the others were shot.

The attack was carried out by terrorists, said police chief Benson Kibue. He did not specify a group. He said it was likely that no more than 10 attackers were involved.

Somalia's president -- the leader of a country familiar with terrorist attacks -- said his country knows "only too well the human costs of violence like this" as he extended prayers to those in Kenya.

"These heartless acts against defenceless civilians, including innocent children, are beyond the pale and cannot be tolerated. We stand shoulder to shoulder with Kenya in its time of grief for these lives lost and the many injured," President Hassan Sheikh Mohamud said.

The gunmen carried AK-47s and wore vests with hand grenades on them, said Manish Turohit, 18, who hid in a parking garage for two hours.

"They just came in and threw a grenade. We were running and they opened fire. They were shouting and firing," he said after marching out of the mall in a line of 15 people who all held their hands in the air.

A local hospital was overwhelmed with the number of wounded being brought in hours after the attack, so they had to divert them to a second facility. Dozens of people were wounded. Officials said Kenyans turned out in droves to donate blood.

The United Nations secretary-general's office said that Ban Ki-moon has spoken with President Uhuru Kenyatta and expressed his concern. British Prime Minister David Cameron also called Kenyatta and offered assistance.

Kenyan authorities said they have thwarted other large-scale attacks targeting public spaces. Kenyan police said in September 2012 they disrupted a major terrorist attack in its final stages of planning, arresting two people with explosive devices and a cache of weapons and ammunition.

Anti-terror Police Unit boss Boniface Mwaniki said vests found were similar to those used in attacks that killed 76 people in Uganda who gathered to watch the soccer World Cup finals on TV in July 2010. Al-Shabaab claimed responsibility for those bombings, saying the attack was in retaliation for Uganda's participation in the African Union's peacekeeping mission in Somalia.

Associated Press reporters Tom Odula in Nairobi, Kenya, Carley Petesch in Johannesburg and Cassandra Vinograd in London contributed to this report.