Claims have been received by the BBC and an Israeli human rights group that Israeli troops have fired on Gaza residents trying to escape the conflict area. Israel has strongly denied the allegations.
BBC journalists in Gaza and Israel have compiled detailed accounts of the claims.
Some Palestinian civilians in Gaza say Israeli forces shot at them as they tried to leave their homes - in some cases bearing white flags.
One testimony heard by the BBC and human rights group B'tselem describes Israeli forces shooting a woman in the head after she stepped out of her house carrying a piece of white cloth, in response to an Israeli loudhailer announcement.
The Israeli military has dismissed the report as "without foundation".
The BBC has spoken to members of another family who say they are trapped in their home by fighting and have been shot at when they tried to leave to replenish dwindling water and food supplies, even during the three-hour humanitarian lull.
Israel is denying access to Gaza for international journalists and human rights monitors, so it is not possible to verify the accounts.
B'tselem said it had been unable to corroborate the testimony it had received, but felt it should be made public.
'Home destroyed'
Munir Shafik al-Najar, of Khouza village in the south-east of the Gaza Strip, told B'tselem and the International Committee of the Red Cross (ICRC) of a series of events on Monday which he said left four members of his extended family dead.
He told the BBC that some 75 members of his extended family had ended up huddled in a house, surrounded by Israeli forces, after troops shelled the area and destroyed his brother's home on Sunday night.
On Monday morning, he said the family heard an announcement over a loudspeaker.
"The Israeli army was saying: 'This is the Israeli Defence Forces, we are asking all the people to leave their homes and go to the school. Ladies first, then men.'
"We decided to send the women first, two by two," he said.
First to step outside was the wife of his cousin, Rawhiya al-Najar, 48.
"The army was about 15 metres (50 feet) away from the house or less. They shot her in the head," he said.
The woman's daughter was shot in the thigh but crawled back inside the house, he said.
For several hours, the family telephoned the Red Crescent, human rights organisations and Palestinian Authority officials in Ramallah in the hope of co-ordinating safe passage to evacuate people injured in the earlier shelling, Mr Najar said.
Several hours later, no help had arrived.
"We decided that's it, we're going to die, we are [going] to run and all die at once," he said.
"When we did that they started shooting with heavy ammunition from a machine gun on top of a tank," he said.
All the adults carried white flags, he said, adding that he was still grasping a piece of white cloth as he spoke over the telephone a day later.
Three of his relatives, Muhammad Salman al-Najar, 54, Ahmad Jum'a al-Najar, 27, and Khalil Hamdan al-Najar, 80, were killed, he said.
The troops "knew this man was an old man," he said, because they were so close.
B'tselem says it is working to corroborate the account.
Similar account
A second family member, Riad Zaki al-Najar, gave the BBC a similar account by telephone.
"They told us you all have to go to the centre of the town, where the school is.
"We put the women first, and we put our children on our shoulders, with white bandanas on their heads.
"When we were walking, with the women first, they saw soldiers and they started to shout to them, to tell them 'we have children, we have children'. They started to shoot us. My aunt was killed with a bullet in her head."
The BBC also spoke to Marwan Abu Rida, a paramedic with the Palestinian Red Crescent, who says he was called to the site at 0810 local time (0610 GMT).
But he says he came under fire as he tried to reach it, and was trapped in a house nearby until 2000 (1800 GMT) because of Israeli shooting.
He said that when he reached the location he found the dead woman, Rawhiya, who appeared to have been shot in the head, as well as the younger woman who was injured.
In a written response to the incident, the Israeli military said: "An initial inquiry into the allegation raised by B'tselem has concluded that the claims are without foundation.
"The IDF goes to great lengths to avoid harming Palestinians uninvolved in combat and reiterates that it is Hamas that chooses to launch its attacks against Israeli towns from within civilian areas."
'Fired upon'
The account bears similarities to another received by B'tselem, from Yusef Abu Hajaj, a resident of Juhar al-Dik, south of Gaza City.
He told B'tselem his mother and sister were shot as they tried to flee their home bearing a white banner, in a group of people including small children.
He said an Israeli tank had fired at their house, and they had heard the Israeli military was urging civilians to leave their homes, so had tried to flee.
The ICRC has repeatedly stressed that it is having difficulty reaching families stranded by the fighting, often including injured people and dead bodies.
Its Gaza spokesman, Iyad Nasser, said ambulance crews were struggling to respond to "tens" of calls from areas they still had not gained sufficient access to.
The head of one such family, Daoud Shtewi, told the BBC by telephone that he and 35 members of his family had been trapped in their home, surrounded by Israeli forces, in Zeitoun, a south-eastern suburb of Gaza City, for 10 days.
"We can't even look through the windows because we get fired on," Mr Shtewi said.
"We tried to get water from the neighbours because our tanks are running dry. We are also running out of food and have been without electricity for more than 12 days.
"My mother and father need medicines for high blood pressure and diabetes. We have run out."
The area, known to house Palestinian militants, has been the scene of some of the heaviest clashes during Israel's operation in Gaza.
It is one of several that Palestinian Red Crescent convoys have been struggling to reach.
Three-hour ceasefire
It was also the place where the ICRC said it found four small children who had waited with their dead mothers, apparently with no food or water, for four days last week.
Mr Shtewi said 17 children - aged between six weeks and 15 years, and six women, were in the house in the west of the neighbourhood.
"We have tried to leave the house during the three-hour humanitarian ceasefire, but we got shot at," he said.
He said the family had repeatedly tried to contact the PRC.
Aid agencies say the Gaza Strip is facing a humanitarian crisis
An ambulance driver with the PRC told the BBC he had received details of a family of 35 people in the location concerned.
But he said it was a closed military zone, that the ambulance workers had not been able to secure co-ordination with the Israeli military to reach it, and were planning to go there as soon as they could secure safe passage with the military.
Israeli military spokesman Jacob Dallal said Hamas was launching rockets from the area in question, and was using civilians' houses - "exactly these types of homes" - to fire rockets from.
"Especially people who try to move out, those could well be - as they have repeatedly been - Hamas people trying to sneak up and fire on the soldiers. If you look from the soldiers' perspective it's exceptionally difficult - you don't know who's behind that door."
He said that Hamas "specifically uses the lull as a time to fire", and Israeli forces fire back if they are fired upon during that period.
And he added that the military was working with international agencies to try to facilitate safe passage for ambulances and the transport of aid amid the fighting.
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