The BBC's Will Ross outside the mall: "Bursts of gunfire can be heard"
At least 59 people were killed and 175 injured in Saturday's attack on a Nairobi shopping centre, Kenya's President Uhuru Kenyatta has said.
More than 1,000 people were brought out of the Westgate shopping centre and the rescue operation continues, Mr Kenyatta said at a news conference on Sunday.
Between 10 and 15 attackers - thought to be militants from the Somali al-Shabab movement - are still inside.
Some civilians are still trapped, either as hostages or in hiding.
'Appalling brutality'
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At the scene
Will RossBBC News, Nairobi
It is extremely tense here as people wait to learn the full horror of this attack. That will only be known once the entire building has been secured.
Trucks full of Kenyan soldiers have been driving towards the mall. Eyewitnesses saw some of those troops entering the building in an effort to end this siege.
But with the government confirming that hostages are still being held, any move will carry great risk. Here, just 200m from the shopping mall, the Red Cross has set up a medical centre to help the injured, including soldiers, who are being rushed out in ambulances. Relatives are registering the names of their missing loved ones and waiting anxiously for news.
An attack by al-Shabab had long been feared. This shopping centre was considered a prime target partly because it is frequented by many different nationalities. This horrific attack has sent this city into shock.
"The criminals are now located in one place within the building," Mr Kenyatta said.
"With the professionals on site, we have as good a chance to neutralise the terrorists as we could hope for."
He thanked those who had helped with rescue and relief efforts, and asked other countries not to issue travel advisories against visiting Kenya.
Mr Kenyatta's nephew and his fiancee were among the dead, the president said.
The UK Foreign Office has confirmed that three Britons have been killed, and says the number is likely to rise.
French, Chinese, Ghanaian and Canadian citizens are also among the foreigners confirmed killed, along with a dual Australian-British national.
British Prime Minister David Cameron called it "an absolutely sickening and despicable attack of appalling brutality".
There is a heavy military presence both in and around the shopping centre. Sporadic gunfire can be heard from inside.
There are reports that the gunmen are currently holed up in a supermarket. Mr Kenyatta said there were reports of women among the attackers but these could not be confirmed.
Minister Joe Lenku says the operation at the shopping centre is "very, very delicate"
BBC security correspondent Frank Gardner says a security source told him that at least one of the attackers was a woman who appeared to have a leadership role.
The Somali militant group al-Shabab says it carried out the attack in response to Kenyan military operations in Somalia.
There are about 4,000 Kenyan troops in the south of Somalia, where they have been fighting the militants since 2011.
'Watching and monitoring'
Kenyan officials said "major operations" were under way, with police and soldiers preparing an apparent bid to bring an end to the stand-off.
The BBC's Will Ross at the scene said it would be extremely difficult for the military do a quick raid on the building because of all the people inside.
Al-Shabab has claimed there are at least 36 hostages, but this cannot be independently confirmed.
Our correspondent says the full extent of the attack will not be known until the military is back in control.
Meanwhile, people are still escaping from the building.
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Confirmed victims
- Ruhila Adatia-Sood, Kenyan radio host
- Ghanaian poet Kofi Awoonor, 78
- President Uhuru Kenyatta's nephew, Mbugua Mwangi and fiancee Rosemary Wahito
- Canadian diplomat Annemarie Desloges, 29
- Two French nationals
- One Australian
- Three Britons
- One Chinese woman
- A second Canadian national
Cecile Ndwiga got out on Sunday morning, saying she had been hiding under a vehicle in the basement car park but could not leave earlier because "the shootout was all over - left, right".
The authorities have asked journalists to exercise caution when reporting military developments because the gunmen might be monitoring the media.
The authorities are also appealing for Kenyans to donate blood. Big queues have formed at a Nairobi donation centre.
The attack began at about 12:00 local time (09:00 GMT) on Saturday, when the militants entered the Westgate centre, throwing grenades and firing automatic weapons. A children's day was being held at the time - children are among those reported killed.
Witnesses report seeing many bodies strewn round tables of unfinished fast food - with pop music left playing in the background.
By Sunday afternoon, the security forces had the attackers pinned down in one corner of the shopping centre, officials said
Military and police are out in force at the Westgate centre, trying to capture the gunmen
Local residents watched from a safe distance as the siege continued for a second day
Funerals have already begun taking place for some of the dead - including Rehmad Mehbub, an 18-year-old Muslim killed in crossfire between police and attackers
Thousands gave blood at donation centres across Nairobi on Sunday
Some witnesses said the militants told Muslims to leave and said non-Muslims would be targeted.
"They came and said: 'If you are Muslim, stand up. We've come to rescue you'," said Elijah Lamau.
Manager: "Bullets were running over my head... I crawled into my office and I locked myself inside."
He said the Muslims left with their hands up, and then the gunmen shot two people.
The US State Department said it had reports that American citizens were injured in what it called "a senseless act of violence".
Prominent Ghanaian poet Kofi Awoonor - who was attending a literary festival in Nairobi - also died, as did a Chinese woman.
UN Secretary General Ban Ki-moon condemned the attack and said one of the victims was a retired staff member from the UN children's agency Unicef.
Security experts are reported to have warned that the Israeli-owned complex was in danger of being subjected to a terror attack.
Al-Shabab, which is part of the al-Qaeda network, has repeatedly threatened attacks on Kenyan soil if Nairobi did not pull its troops out of Somalia.
This is one of the worst incidents in Kenya since the attack on the US embassy in August 1998.
Footage from inside the mall shows the aftermath of the shoot-out
Pakistani Christians held protests following the Peshawar attacks
Islamists have been blamed for attacks on Christians in the past
Grieving relatives gathered to identify their relatives
The Pakistani Christian minority feels vulnerable to militant attacks
Security has been strengthened outside a number of Pakistani Christian churches following Sunday's attack
