A patient waited 62 hours for an ambulance, while four trusts took more than 24 hours to respond to 999 calls, new figures have shown.
The longest delays in the UK were recorded by Welsh Ambulance Service, which kept four patients waiting for more than 50 hours.
A spokesman said the figures were "not typical" and "represent the extreme end of the waiting time spectrum".
The Patients Association said they were "extremely concerning".
Between June 2017 and June 2018, ambulances from four services took 24 hours to reach patients, including some with breathing and mental health problems.
The trusts said the longest waits were for "less serious calls", and they had to prioritise responding to people in life-threatening or urgent conditions.
Most ambulance services also reported achieving the national target of responding to the most serious type of call in an average of eight minutes or less.
Lucy Watson, from the Patients Association, said: "Everybody should be getting the services that they need.
"We know that demand has gone up on all health services as our population is getting older, and we need to see the level of investment increasing so our ambulances can respond in a timely way."
Caroline Hardaker's mother Sylvia, 79, lay on paving stones in her back garden for three-and-a-half hours after falling and breaking her hip in High Wycombe.
She said: "I think I rang six times in the end, and each time they said they would have a clinician call back and then they didn't.
"It was so frustrating, just thinking 'how long are they going to take'? And my mum was going into shock, her arms were shaking. Her arm had gone numb.
"She was obviously cold because she was lying on a pavement.
"The ambulance and hospital staff have been fantastic – its not their fault, the whole system is breaking down.
"When I was a child you were told it would be a maximum of eight minutes for an ambulance, but three-and-a-half hours is completely unacceptable."
Paul Jefferies, from South Central Ambulance Service, which covers High Wycombe, said if patients experienced delays it was because "higher categories of calls took priority".
Stephen Clinton, assistant director of operations for Welsh Ambulance Service, said: "We fully accept that a number of patients waited far longer than anyone would like.
"That said, these figures represent the extreme end of the waiting time spectrum and are neither typical nor do they explain the circumstances of these individual cases."
He said in some of the cases the patients were already in the care of medical teams, and others were affected by extreme weather conditions.
The service did not provide details of the four patients who waited more than 50 hours.
But the longest three calls were in the second-most serious "amber" category, classified as "patients who may need treatment at scene or taking quickly to health facility".
The remaining call was rated "green", a classification used for "less urgent calls".
The new figures, obtained by the BBC after a Freedom of Information request, also show that between 2015 and 2017, the total number of calls received increased by 15%.
In 2015, UK ambulance services received 8,892,346 calls, which rose to 9,891,559 in 2016 and 10,242,507 in 2017.
An Association of Ambulance Chief Executives spokesman said resources had been stretched by "an exceptionally long and busy winter".
An NHS Improvement spokesman said it had recently introduced an ambulance response programme to help services cope with the increasing demand.
He added: "An additional £36 million of funding recently announced will boost paramedic crews and improve the quality of NHS ambulance fleets."