(CNN) Judges on the UK Court of Appeal once again ruled against 23-month-old Alfie Evans' family.

Wednesday's ruling rejected new arguments intended to overturn a decision by the High Court on Tuesday that prevented the terminally ill toddler from leaving Britain for medical treatment, said Roger Kiska, a lawyer with Christian Legal Center and part of the legal team representing Alfie's parents.

"We have the right to appeal to the Supreme Court and the European Court," Kiska told CNN. "We could seek new medical evidence showing that his condition is improving." Kiska will be discussing future steps with Alfie's parents, Tom Evans and Kate James, who remained in Liverpool at Alfie's hospital bedside.

Alfie, admitted to Alder Hey Hospital in December 2016, was diagnosed with a neurodegenerative disease associated with severe epilepsy and has been in a semivegetative state for more than a year. During that time, he has been kept alive by artificial ventilation in the critical care unit.

"Alfie has received the full support of Alder Hey's medical and nursing teams since being admitted," the specialist children's hospital said in a statement. "Sadly he has remained unresponsive to treatment and his condition has rapidly declined."

People protest outside Alder Hey Hospital where 23-month-old Alfie Evans is being cared for in Liverpool, England.

Based on "extensive damage" in the child's brain, the hospital recommended to the child's parents that active treatment be stopped.

Disagreement between the hospital and his parents resulted in a referral of Alfie's case to the Family Division of the UK High Court for a judge to rule on whether active treatment is in Alfie's best interests.

When the court did not rule in their favor, the parents appealed the decision, and after that appeal was lost, the case was referred to the Supreme Court.

Meanwhile, Alfie's plight -- legally and medically -- has stirred the British public, and hundreds of thousands have signed petitions supporting him.

Papal support

While the case played out in the legal arena, the hospital continued to provide Alfie with artificial ventilation and assisted feeding, but his doctors said continued treatment was futile and speculated that, if they were to withdraw Alfie's ventilation support, the child would shortly die.

Last week, Alfie's father traveled to Rome to meet Pope Francis, who had expressed his support for their cause.

At that time, it was suggested that Alfie could be transported to Rome's Bambino Gesu Pediatric Hospital, a Vatican hospital and the main pediatric hospital serving southern Italy, for treatment.

However, a subsequent visit to Alfie and consultation with his doctors led the Roman doctors to conclude that the child's condition is irreversible and untreatable, according to a statement from Alder Hey.

Based in part on evidence supplied by the hospital, the supreme court dismissed Alfie's case. High Court Justice Anthony Hayden rejected the plan to take Alfie to Rome and said the order to end Alfie's life support should commence at 9 p.m. Monday.

In a last-ditch effort, Alfie's parents appealed to the European Court of Human Rights, which rejected their application on Monday, declaring the complaint "inadmissible."

At 9:17 p.m. Monday, ventilation support was withdrawn.

Breathing independently

Contrary to all the expectations of the doctors, Alfie continued to breathe independently, according to a statement from Christian Concern, a sister organization of the Christian Legal Centre, which is representing the parents.

Tom Evans also posted a video on Facebook in which he said that his son had been breathing for himself and that the doctors were refusing to give him oxygen.

Tom Evans, father of Alfie Evans, speaks to media outside Alder Hey Children's Hospital.

For hours, Alfie continued to breathe unaided, with hospital staff refusing to help with either ventilation or hydration despite his coughing, according to the Christian Concern statement and Alfie's father.

About 2 a.m. Tuesday, Alder Hey hospital staff provided some hydration and oxygen support, his mother reported on Facebook.

The Italian government granted Italian citizenship to Alfie on Tuesday and applied to the foreign secretary for permission to intervene to save Alfie's life.

Once again, the courts had to decide, in an urgent hearing on Tuesday, whether this would be permitted, and it ruled to ban travel.

Thomas Evans meets with Pope Francis.

In Wednesday's appeal hearing, Alfie's legal team argued that the hospital's original end-of-life plan was based on suppositions that the toddler needed oxygen to survive, Kiska said. The fact that Alfie lived on even without ventilation support was a material change in circumstance and served as grounds for renewing the matter.

It is in Alfie's best interest to travel to Italy, where doctors are ready to care for him, the legal team argued in court.

"We're still holding out hope because Alfie continues to fight," said Kiska. Since the toddler has been "denied nutrition" for more than 24 hours, he said, "we're going against the clock."

A representative of the Italian Embassy attended the hearing, while an air ambulance waited outside Alder Hey Hospital to fly Alfie to Italy.

Andrea Williams, chief executive of Christian Concern, said, "it cannot be in Alfie's best interests to be left in Alder Hey where the court order is to let him die. Alfie's best interests would be served by allowing him to travel to Italy."

Williams said the "common sense approach" would be for Hayden to discharge all the court orders which detain Alfie.

Having failed to gain the court's permission to fly his son to Italy, Tom Evans told British media on Thursday morning that he will be speaking to the doctors at Alder Hey about bringing Alfie home.

Professor Dominic Wilkinson, a consultant neonatologist at the John Radcliffe Hospital and director of Medical Ethics at the Oxford Uehiro Centre for Practical Ethics at the University of Oxford, said, "In Alfie's case, none of the foreign experts have offered any new treatment or any new outlook for Alfie. The Italian specialists have apparently indicated that they believe that his care should be palliative."

Wilkinson added that hospitals and courts need to follow an ethical framework, and in this case, they have decided treatment should no longer be provided, despite the parents' wishes.

"Sometimes, the sad fact is that parents do not know what is best for their child," Wilkinson said. "They are led by their grief and their sadness, their understandable desire to hold on to their child, to request treatment that will not and cannot help."