There needs to be a "revolution" in societal attitudes towards organ donation so more lives can be saved, the NHS Blood and Transplant (NHS BT) has said.
The proportion of families who agree to organ donation after the death of a family member remains "stubbornly low", NHS BT spokeswoman Sally Johnson said.
Figures show that during 2013/14, more than four in 10 families approached about organ donation said no to donating a loved one's organs.
"Family refusal is our biggest problem and it's sad we lag so far behind some other countries in terms of consent/authorisation rates to donation," she added.
More people are donating their organs in Britain than ever, according to health officials
The last financial year was a "record year" for organ donation and transplantation in the UK, NHS Blood and Transplant (NHS BT) has said.
In 2013/14 there were 4,655 transplants carried out - a 10% rise on the previous year, a spokeswoman said.
New figures, published by the authority to mark National Transplant Week, show that almost one in four of the transplants were organs from "living donors".
The remainder of organ donations came after a death.
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NHS Blood and Transplant (NHSBT) officials have offered their "heartfelt thanks" to the families who donated their deceased loved one's organs and "helped transform the lives of others".
Sally Johnson, director of organ donation and transplantation at NHSBT, said.
Last year we set out our aspiration to match the best countries in the world for organ donation and transplantation, and if we are going to achieve this we will need to see a revolution in attitudes in society towards donation.
The increase in donors reflects increasing support in hospitals to refer potential donors to us and more families being approached, but there has been little change in our attitude to donation over the last few years. It's still not something we would all be proud to do.
A record number of people have donated organs after death in the UK, health officials announced.
Last year organs were taken from 1,323 people who died - a 13.7% rise on the previous year, NHS Blood and Transplant (NHSBT) said.
The organs were given to 3,500 people, a spokeswoman for the body, which is responsible for organ donations in the health service, said.
The figures for 2013 also show that 58.6% of bereaved families who were asked to donate their loved one's organs did so - a rise from 56.5% the previous year.
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My heart goes out to all the families that find themselves in the position where they have to make a decision about organ donation.
In particular, the decision to donate on behalf of a young child is really tough. But organ donation saves lives and donor families talk with pride of the difficult but ultimately rewarding decision to donate their loved one's organs. Knowing that they have helped save another family from a terrible experience, often helps them deal with their grief.
There are around 10,000 people in the UK in need of a transplant and some of these are young children. In order to save more lives, we need more people to join the organ donor register.
A five-week-old baby has become the youngest organ donor in Britain according to a report in the Sunday Times.
The child died earlier this year from heart failure and their kidneys were transplanted to 22-year-old Samira Kauser.
The paper claims that younger donors are being sought due to the shortage of adult organ donors.
A Midlands doctor is calling for a radical approach to increase the number of black and Asian people donating their organs.
According to Dr Adnan Sharif, a kidney consultant at the Queen Elizabeth Hospital in Birmingham, the UK should consider a scheme where priority for surgery is given to people already on the organ donor register.
He says there has been a 'huge failure' to boost organ donation rates among Asian and other ethnic minority groups.