NAIROBI — Gunmen stormed a popular high-end shopping mall in the Kenyan capital Saturday afternoon, lobbing grenades and firing weapons in an attack that left at least 39 people dead and more than 150 injured, Kenyan officials said.

As of early Sunday, more than 12 hours after the initial assault, the attackers, strapped with grenades and wielding machine guns and AK-47 rifles, remained holed up with scores of hostages in a supermarket within the Westgate Premier Shopping Mall, exchanging gunfire with Kenyan police and soldiers.

Al-Shabab, a Somali militia linked to al-Qaeda, asserted responsibility for the assault in numerous tweets using its official Twitter handle, @HSM_Press. The militia said it was retaliating for Kenya sending troops to fight in neighboring Somalia, where it remains a key military actor. “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,” the militia said in one tweet.

The dead and injured included young and old, Kenyans and foreigners, according to witnesses and a U.S. State Department official familiar with the situation, who spoke on the condition of anonymity because the official was not authorized to speak publicly. No Americans were believed to be among the dead, the official said. Four U.S. citizens were reported injured, the Associated Press said.

Several children were reported killed or injured.

Annamaria Watrin, an American aid worker from Minnesota, said a friend and his 13-year-old daughter had gone to the mall for a birthday party. “As they went to park their car, she saw five gunmen pop out. They shot her dad. He died,” Watrin said. The girl was injured. Watrin said the girl spent a couple hours huddled in the car before Kenyan security agents could evacuate her into an ambulance.

The assault was the deadliest terrorist attack in this East Africa nation since al-Qaeda operatives staged twin bombings of the U.S. embassies in Nairobi and Dar es Salaam, Tanzania, in 1998, killing more than 200 people in Kenya alone. Al-Shabab has staged numerous smaller attacks in the country since the government sent troops to Somalia in October 2011 to fight the militia. Most of those assaults targeted bus stations and churches, but never areas frequented by Westerners and wealthy Kenyans. The tourism industry is Kenya’s second-largest source of foreign exchange, and dozens of Western aid agencies and businesses are based in the country.

Now, Saturday’s attack has appeared to usher in a new war on its soil for Kenya, long a bulwark of stability in the region and a key U.S. ally in the fight against terrorism.

The militia also orchestrated twin bombings in Kampala, Uganda, during the World Cup in July 2010, killing more than 70 people. That attack, the militia said, was in retaliation for the participation of Ugandan soldiers in an African force sent to protect Somalia’s government.

In a nationally televised address, President Uhuru Kenyatta vowed to hunt down the perpetrators.