Margaret Mary Price, an Irish gypsy, has lived with her six children in caravans at Prince Charles Gardens, Llanelli, since May this year.
The council say they have done all they are legally obliged to do by offering the family a home on 5 July this year, and are now trying to evict the three caravans from Prince Charles Gardens.
But the family say they have nowhere else to go, and claim that the council is being 'unfair and irrational' in expecting them to live at a permanent address.
Stephen Knafler, for Mrs Price, explained the situation to Mr justice Newman at the high court.
'Mrs Price is an Irish traveller by descent and culture who has travelled from place to place for her entire life in a caravan.'
He told how the family moved onto Prince Charles Gardens in May on the understanding their presence would be 'tolerated' until 7 July.
But he said that 'unfortunately Mrs Price has been unable to leave', because all official gypsy sites in the area are filled to capacity.
'The council have not identified any piece of land where Mrs Price and her family could pitch their caravans and where their presence would be tolerated for a reasonable, or any period of time.
'There's no evidence and no suggestion that Mrs Price and her family are causing any kind of nuisance.
'Although the area is called Prince Charles Gardens, it does not comprise gardens at all but is an area of wasteland at least half a mile from the nearest houses,' Mr Knafler told the court.
He concluded that, under the Housing Act 1996, the council was obliged to let the family stay where they are, or provide 'suitable' alternative accommodation for their caravans.
He challenged as 'irrational and unlawful' the council's stance that the council has performed its statutory duty by offering the family a permanent house to live in.
Carmarthenshire CC is hotly resisting Mrs Price's judicial review challenge and, after a day of legal argument, Mr justice Newman reserved his decision until a later date.
STRAND NEWS SERVICE