"If there is a little hope that there is life, then it's worth fighting for," the family's attorney, Chauncey Henry said.
The 23-year-old Monroe College business student has been on life support at Montefiore Hospital in the Bronx for several weeks after suffering a major stroke during surgery.
"My sister came to this hospital driving a car. Literally. She was perfectly fine," her sister, Kay Ebanks, said.
Her family won a temporary restraining order last week against the hospital to keep her on life support.
"And we have a team of medical professionals that have evaluated and assessed Amber's medical records and are of a very strong opinion that her life is worth fighting for. Montefiore on the other had wants to pull the plug. But there won't be any plug-pulling today," Henry said.
On Monday, the court heard arguments from those experts, specifically as to whether Montefiore must continue to provide life support and food and nutrition to Amber during a one-to-two-week transfer period to a facility called New Beginnings, a facility on Long Island that has agreed to care for Ebanks as long as her family would like.
"Everybody needs hope. You can't just give up. Can't just take them off life support when she needs more time," New Beginnings Founder Allyson Scerri said.
Amber's family members, who are mostly in Jamaica, say they need more time with her.
Her dad is in the process of securing a Visa to see her and New Beginnings says he can do that there.
Amber's sister Kay says the hospital suggested family members say goodbye over the phone.
"They are some of the cruelest people I have ever known," Kay Ebanks said.
Montefiore said in a statement that it couldn't comment due to patient privacy laws.