Cumbria Police said two females and a male were injured at the Fudstock festival in Barrow-in-Furness on Saturday night (August 28).
One of the females remained in a critical condition in hospital, police said on Sunday (August 29).
The Health and Safety Executive and Cumbria Police have launched a joint investigation after the incident at Cavendish Park.
Following the incident, organisers posted a statement on their Facebook page, saying: "Thank you to everyone who joined us at Fudstock 2021.
"Unfortunately, there was a serious incident at the end of the night involving three people on one of the fairground rides.
"Our thoughts and best wishes are with those involved, as we await further news.
"We appreciate this is upsetting and we'd ask people to be kind and not speculate out of respect to the families and friends."
Fudstock is an annual community music festival launched in 2018 to commemorate the life of Richard Thorne - known as Fud –-who died of bowel cancer at the age of 38.
The not-for-profit festival raises funds for St Mary's Hospice in Ulverston, which cared for Mr Thorne in the last weeks of his life.
The hospice released a statement on social media offering support to the event’s organisers, fundraising staff and volunteers who were at the festival.
Val Stangoe, Chief Executive at the Hospice, said: "The team who organise Fudstock have supported the hospice over a number of years.
"We will do whatever we can to support them at this difficult time.”
Police were called to the scene at 8.30pm on Saturday by North West Ambulance Service.
The three injured people were taken to Furness General Hospital.
A force spokesman said: "Two of the people were reported to have been on a ride at the time that the incident occurred.
"The injured male and one female have since been released from hospital.
"The third female has been transferred to Royal Preston Hospital where she remains in a critical condition."
Furness Coastguard helped the ambulance service transport the injured people.
A spokesman said: “Furness Coastguard were paged tonight (Saturday) at 9.32pm to assist North West Ambulance Service NHS Trust and Coastguard Rescue Helicopter 936 with casualty evacuations from Furness General Hospital following a major incident."
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