Three UK residents held by the US at Guantanamo Bay since 2003 are to appeal against a court's refusal to force the government to request their release.
Mr Deghayes's family have backing from their local MP
Judges indicated the Foreign Office was correct to say it could not provide diplomatic protection to non-citizens.
But the men's relatives, many of whom are British, said the High Court decision breached human rights laws.
The sister of detainee Omar Deghayes, 37 and from Brighton, described the decision as a "bitter defeat".
Amani Deghayes said: "Guantanamo has been condemned by so many independent organisations, it's just not good enough for the UK government to say that they can't even make a plea."
Her other brother Abubaker described the ruling as a "setback" but highlighted the fact the judges found "powerful submissions" had been raised.
"The judges recognised and stated in their ruling that there is a strong case and we are going to appeal against the decision," he said.
"Time is not on our side because my brother is suffering and every minute counts but I haven't lost faith in the judicial system and I think it will work out to be just."
In seeking a judicial review of the government's decision not to petition for the men's release, their lawyers contended they were entitled to similar government help to that received by UK citizens freed from Guantanamo in 2004 and 2005.
They argued there was evidence Libyan Mr Deghayes, Iraq-born Bisher al-Rawi and Jordanian Jamil el-Banna have "suffered inhuman and degrading treatment" at the camp.
The government is making efforts to secure the closure of Guantanamo Bay, in Cuba.
In March, United Nations human rights investigators called for the camp's immediate closure.
But the US government has been steadfast in its defence of the camp, established to hold suspected terrorists, and says inmates are treated humanely.
The families' deny the three UK residents detained at Guantanamo have links to terrorism.
Their solicitor Gareth Peirce said the UK's position on men "marooned in conditions of torture" was a disappointment.
She said there would be an appeal as there were "strong arguments" on which to challenge the ruling, including under the Race Relations Act.
"This is not an area where the government is entitled to discriminate and it has discriminated between nationals and residents," she said.
Mr el-Banna's wife Sabah was present in court with two of their five children for the ruling.
"We must continue and we will appeal," she said.
Omar Deghayes came to the UK in the 1980s
Lawyers for Mr al-Rawi had told the court he helped British intelligence in the past.
And the government has said it is to make a separate "security-related request" on behalf of the 38-year-old.
Amnesty International UK Director Kate Allen said the government's reluctance to act on behalf of long-term residents of the UK was "shameful".
She added: "Instead of propping up the government's policy of picking and choosing which of the UK residents held prisoner at Guantanamo it was willing to stand up for, the High Court could have sent out the message that they all need the government's help."
Labour MP Des Turner, whose constituency covers Mr Deghayes' former home in Brighton, said there remained a "moral pressure" on the government to act.
The Foreign Office welcomed the ruling.
It said Mr al-Rawi and Mr Deghayes, who had lived in the UK since the 1980s, had each made specific representations to Foreign Secretary Jack Straw and it was assessed there were grounds to act in the one case.