Seven more people died on Saturday, according to the British Ministry of Defence, and the total number of deaths in and around the airport has reached at least 20 since the Taliban seized the city last Sunday.
One of the seven people crushed to death in a stampede outside the gates of the airport was a two-year-old girl.
The grief-stricken mother said that her family – husband, three sisters, elderly parents, and cousin – will abandon their attempts to flee Afghanistan following the death of their toddler.
Elsewhere, a woman who made it out of the country gave birth aboard a US Air Force C-17 aircraft, the US military said.
While on the way to the Ramstein Air Base in Germany, which is to temporarily host the refugees on their way to the US, the Afghan woman experienced complications while giving birth as a result of the reduced air pressure inside the aircraft.
The US military’s Air Mobility Command tweeted: “The aircraft commander decided to descend in altitude to increase air pressure in the aircraft, which helped stabilise and save the mother’s life.”
On arrival at Ramstein medics boarded the plane and delivered the baby in the aircraft’s cargo bay. Images show the woman being carried out from the plane on a stretcher.
“The baby girl and mother were transported to a nearby medical facility and are in good condition,” the US military said.
The MoD said almost 6,000 people have been evacuated as part of the UK rescue mission in Kabul.
Those repatriated under Operation Pitting include embassy staff, British nationals, those eligible under the Afghan Relocation and Assistance Policy (Arap) programme and a small number of nationals from partner nations.
The evacuation is being supported by 1,000 British troops on the ground – including Paras from 16 Air Assault Brigade – as well as other Whitehall staff.
Brig Dan Blanchford, the most senior UK military officer on the ground in Kabul, said British armed forces personnel had “witnessed some harrowing scenes”, with at least seven Afghan civilians confirmed to have died outside the airfield gates in the chaotic crowds.
In total, 5,725 people have been repatriated since the mission began on 13 August, with 3,100 of them Afghan individuals and their families.
The MoD said defence secretary Ben Wallace had a phone call with his US counterpart, Lloyd Austin, to discuss allied efforts in the Afghan capital.
The MoD said the evacuation process would “run as long as the security situation allows in joint coordination with our US partners", with "no firm date set" for the end of civilian flights.
Brigadier Blanchford, commander Joint Forces Operations, said the armed forces had also been supplying aid – including food, nappies and baby milk – as part of their efforts to get people to the UK.
UK armed forces minister James Heappey said the Taliban were now marshalling people into queues at Hamid Karzai International Airport, making the process faster for those hoping to leave. The UK has airlifted out more than 1,700 people in the past 24 hours, he said.
British officer Lt Col Justin Baker, of 16 Air Assault Brigade, said that troops are dealing with an unprecedented, complex situation in Kabul.
Thousands of people are struggling through the brutal heat, distress, and unsanitary conditions outside the airport in a desperate bid to escape the Taliban’s rise to power.
Images have shown soldiers spraying people with water from a hose, and giving out bottles of water to those who have been waiting for a space on a flight for days with no guarantee that they will make it out.
Afghan people in tightly-packed crowds are also seen handing their babies and children to soldiers in the hope that they would be safer, and that the youngsters can will be able to leave the country.
US president Joe Biden’s national security adviser, Jake Sullivan, said that US troops have evacuated 3,900 people in the past 24 hours, and that non-US military flights evacuated a similar number of people over the same period.
Mr Sullivan has not ruled out the possibility of the Biden administration deploying more troops, in addition to the 6,000 already deployed, to help evacuate more people.
He said that Mr Biden asks his defence chiefs “every single day” whether they need more troops or other resources to ensure safe evacuations of Americans and non-US citizens from Kabul airport.
In a speech on Sunday, Mr Biden repeated his pledge that “any American who wants to get home, will get home”, and that the US is working with more than two dozen countries on the evacuations.
He said that more than 11,000 people have been evacuated from Kabul airport this weekend.
But western allies have stressed that not all people who need to be rescued will be able to be saved by the planned deadline of 31 August for all evacuations. Several nations, including the UK, are calling for this deadline to be extended.
And time has become more critical as the US fears that there are “acute” potential threats posed by groups such as Isis and al-Qaeda. The Taliban has since claimed that it is not affiliated with al-Qaeda, and that the group has no presence in Afghanistan.
Washington’s struggle to evacuate people amid the mass attempted exodus has also been evident by the Pentagon calling on six US commercial airlines to use 18 of their civilian aircrafts to carry people to the US from transit locations.
Amid the evacuation difficulties, former British PM Tony Blair made a series of extraordinary remarks slamming Britain and the US for their withdrawal of troops from Afghanistan.
On Saturday, in his first comments on the situation since the Taliban’s takeover, he said that the “abandonment” of the country by western allies is “dangerous and unnecessary”.
He added that Mr Biden withdrew troops “in obedience to an imbecilic political slogan about ending ‘the forever wars’”.
Mr Biden and PM Boris Johnson will meet virtually with other G7 leaders on Tuesday to discuss the ongoing situation in Afghanistan.
Mr Johnson is expected to persuade Mr Biden to keep troops in Afghanistan beyond 31 August to help evacuate more people.