Zoe Lightfoot has infantile neuroaxonal dystrophy (INAD), a condition which means an affected child usually dies between the ages of five and 10.
She was diagnosed in April after an MRI brain scan in January revealed significant brain damage and cerebellar atrophy.
The part of the brain that affects balance and co-ordination, speech and swallowing had formed normally as a baby, but then shrivelled.
Her parents Christine Hamshere and Steven Lightfoot, of Bramley, who are expecting another child in March, have raised more than Â£16,000 of a Â£20,000 target to take Zoe to America to participate in a clinical trial with California-based Retrotope.
They hope this would halt and possibly reverse some of her symptoms.
Miss Hamshere, 41, said: “As soon as she was diagnosed, I really struggled. We sent close friends links to information and said, ‘Brace yourselves, this is what’s going on’.
“It was awful. Zoe was sat on my lap. We didn’t think they had any results, we just wandered in thinking, ‘This will be quick’.”
She added: “I just couldn’t take it.”
According to the Great Ormond Street Hospital, which specialises in care for children, INAD is a rare inherited disorder affecting the nerve axons which are responsible for conducting messages in the brain and other parts of the body.
It causes a progressive loss of vision and of physical and mental skills.
Since the diagnosis, Zoe has lost mobility and her speech.
Swallowing has also deteriorated, and she has some leg stiffness.
Despite this, according to the family’s online fundraising campaign, she is “still a bright bubbly fun loving beautiful girl.
“She has a great sense of humour, an infectious laugh, and still tries to wave and blow kisses. She loves bouncing along to songs and interaction with others, cuddling the neighbour’s dog and her toy rabbits.
“She has a good understanding of what is going on around her, and loves it when you copy her actions and sounds, especially drumming. She has a smile that lights up the room. We are desperate to keep her this way for as long as we can.”
The study drug in the trial is a stabilised fatty acid that protects cells from being broken down, the family said.
Zoe will need to make four trips in a year as part of the trial, and then potentially more after that. The first visit is at the start of December, and the second will be in January 2019.
Any funds raised that are not spent on the trial will go towards adapting the family’s house for Zoe, transporting her to school, and supporting all the charities that are helping the family, such as Martin House Children’s Hospice, Special Needs And Parent Support Yorkshire, the Little Hiccups support group and Leeds Mencap.
Visit the fundraising page at https://www.gofundme.com/zoes-inad-battle-fund