The case of Mary Flynn came to court as hundreds of residents, bolstered by supporters, barricaded themselves inside Dale Farm in Basildon, Essex, to resist the mass clearance of the site.
Mrs Flynn, 72, suffers breathing problems and uses an electric nebuliser and has been a crucial protagonist in the bid to stop the clearance.
Today her counsel, Stephen Cottle, said her condition continued to deteriorate and it would be "disproportionate" to evict her.
Basildon Council is due to carry out the clearance of the site following a decade-long legal fight over unauthorised development of the former scrapyard.
It involves 51 unauthorised plots on six acres, home to up to 400 people.
Mrs Flynn is asking Lord Justice Pill for permission to appeal against a High Court judge's refusal last month to grant a temporary injunction to halt or delay the evictions.
Mr Cottle argued today that the enforcement notice which had led to eviction proceedings against Mrs Flynn did not require her eviction but only the taking up of hardcore on the land.
He told the judge: "She is too frail, her medical condition is too pronounced, too severe for her to be evicted for failing to take up hardcore on her plot."
Mr Cottle apologised to the judge for "rushing" to court today "at the 11th hour and 59th minute" but said the "unseemly rush" could not be avoided owing to problems with funding the case.
Basildon council vowed to go ahead with evicting the families after talks with the residents broke down.
An estimated 200 residents and supporters barricaded themselves inside Dale Farm in Basildon, Essex, as bailiffs prepared to clear the former scrapyard.
The travellers recently lost a decade-long legal fight over unauthorised development on 51 plots on half of the six-acre site.
Basildon Council leader Tony Ball said bailiffs would launch the operation to clear the site later today in a safe manner.
"I am very disappointed that we have come to where we are today," he said.
"I would much rather that we reached a much more peaceful solution, and I am still hoping that once our bailiffs will move on site they will be received in a peaceful manner and that the operation will go ahead as we would like it to proceed."
He added: "Our operatives, when they begin the site clearance, which will be today, will do everything they can to make sure it's done in a safe and professional manner."
Mr Ball said the operation to clear the site could last as long as six to eight weeks.
The council delayed starting the clearance process this morning after the residents requested talks.
But officials called off the meeting once they established that the travellers were seeking to delay the eviction until November 22, when a planning appeal on a nearby site will be heard.
Mr Ball said: "Rightly, the operations team rejected that and cancelled any possible meeting.
"It is unfortunate because we were hoping that if one family had moved off because of that contact, then it was worth waiting for the operation to begin."
He added: "This is about the legality of the situation. The criminal law has been broken. It is about fairness for all, about getting planning permission first, as we all must do."
Travellers inside Dale Farm reacted angrily to claims that only activists remain inside the wire.
Kathleen McCarthy said: "The elderly and sick are here and we will stay. That bluff won't work with us."
Basildon Council said it had offered accommodation to all of the families affected.
Mr Ball said he understood that the travellers on the illegal site were outnumbered by the 50 to 60 supporters protesting against the eviction.
He said: "My personal concern is the intervention of outsiders, who maybe have their own agendas and, again, in my view, not the interests of the travellers at their heart, certainly raises the risk of there being a disturbance."
Essex Police mounted a major operation to prevent violence breaking out at the site. Superintendent Trevor Roe said no arrests were made by 12.15pm.
Campaigner Grattan Puxon condemned the council's approach to the cancelled meeting.
He said: "They only wanted to talk about peaceful eviction, not an alternative site for the homeless, so as far as we're concerned negotiations are off and we're preparing for the bailiffs to move in."
The protesters erected reinforced barricades and built a new wall, and some even chained themselves to obstacles in an attempt to keep the teams of bailiffs out using "peaceful resistance".
Two supporters of the travellers, giving their names as Dean, 29, and Emma, 18, handcuffed themselves to a pole concreted inside a barrel behind the gate to the site.
Dean said: "I have studied what's going on here long and hard and believe when the law is used for wrong, civil disobedience is the only way to oppose it.
"The idea is the bailiffs cannot open this gate without killing us both. We'll sleep here for weeks if we have to."
Paintings of children and a banner calling for "Human rights for Dale Farm" were put up outside the main barricade.
There was also a sign warning about the activists chained to a barrel, stating: "Danger of death. Behind this gate a woman is attached by her neck. If you attempt to open this gate you will kill her."
One protester, called Ruth, who chained herself to a car, said: "There are people here from all over the world and the travellers have made us so welcome.
"It has become increasingly difficult for travellers to find a site in recent years. Planning law is very discretionary and we do not feel it's being used fairly."
Resident Kathleen McCarthy said the travellers feared it was inevitable that they would be forced to leave.
She added: "If they had any human decency this could be stopped, I plead and beg to stop this. Look what these people (the protesters) are doing to save us."
The protesters are demanding that Basildon Council provides 62 plots to temporarily house those made homeless while planning permission is sought for other sites.
Supporters delivered a large amount of supplies, including crisps, bread, tinned fruit, biscuits, peanuts, dried apricots, orange juice and toilet paper, to the site through the main gate this morning.
Meanwhile, elderly traveller Mary Flynn mounted a final legal challenge to the clearance at London's High Court today - but Basildon Council said the hearing would not delay the start of the operation.
The 72-year-old, who has been a crucial protagonist in the High Court battle, suffers from breathing problems and argues that her poor health should have a bearing on the clearance.
Asked how the Prime Minister viewed today's eviction, David Cameron's official spokesman said: "This is obviously a matter for the local council and for the police and we wouldn't want to comment on the operation.
"On the basic case, he was very clear at Prime Minister's Questions last week that the British courts have found that the development at Dale Farm is illegal.
"There has been an unprecedented level of unauthorised development on green belt land that's resulted in years of negotiation at significant cost to the taxpayer.
"As he said at PMQs, everybody in this country has to obey the law, including the law on planning permission."
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