Durham police issued fines to two people - from different households - who travelled together from London to County Durham during lockdown.
The force is currently considering whether to take action against the prime minister's chief aide Dominic Cummings over a similar journey.
Mr Cummings made the 260-mile trip from London to Durham with his wife and four-year-old son.
The two individuals fined by the force travelled to nearby Peterlee.
Mr Cummings has defended his decision to make the journey to Durham, insisting it was legal and within the guidelines.
"The rules made clear that if you are dealing with small children that can be exceptional circumstances," he said at a press conference on Monday.
"And I think that the situation that I was in was exceptional circumstances, and the way that I dealt with it was the least risk to everybody concerned if my wife and I had both been unable to look after our four-year-old."
He has been backed by Prime Minister Boris Johnson, who has said it is time to "move on" from the row and focus on the public's "needs, rather than on a political ding-dong about what one adviser may or may not have done".
The lockdown fines were uncovered in Freedom of Information requests made to police forces around the country before news of Mr Cummings' journey emerged.
The BBC is seeking further details from Durham Constabulary about the two individuals who were fined for travelling from London to Peterlee, about 13 miles east of Durham, on 8 April, a week after Mr Cummings made his trip.
Durham Constabulary is investigating whether Mr Cummings broke lockdown rules with his journey from London to Durham and a subsequent trip to Barnard Castle, about 30 miles from Durham.
Officers have reportedly spoken to the man who told the Guardian and Daily Mirror he had seen Mr Cummings in Barnard Castle, on Easter Sunday, as part of their investigation.
A government spokesperson said "We are confident the police will use their common sense, discretion and experience."
Police powers to enforce protection came in to force in England on 26 March.
By 11 May, 13,445 Fixed Penalty Notices had been handed out in England and 799 in Wales.
The BBC's Freedom of Information requests covered the first two and a half weeks of the regulations, including the Easter Weekend.
In this period, police forces in England issued 3,203 FPNs - an initial fine of £60, halved if you paid within two weeks.
In one case, a man in Leicestershire was issued with an FPN after being stopped on the motorway.
He told officers he was "travelling home to London after visiting a sick relative" in Nottinghamshire.
Another FPN was given to a man who was "taking his motorbike for a ride" 10 miles from his home address.
There is no appeal process for FPNs issued under the coronavirus regulations.
Civil liberties campaigners say there is no consistency in how the rules have been applied and are calling for a review of the "unjust" fines.
Rosalind Comyn, policy and campaigns officer at the rights group Liberty, said: "Such broad police powers and vague government guidance are a recipe for discrimination and injustice, which is why Liberty along with many other groups have been consistently calling for a review.
"It's now clear how unevenly the powers have been applied - particularly when, as recent events show, they don't apply to the very people who wrote them.
"The government urgently needs to pare back the powers, create a right to appeal and review every fine issued."
Kirsty Brimelow QC, a barrister at Doughty Street Chambers, said: "In light of Mr Cummings' actions being upheld by the government as legal and within both law and guidance, there needs to be an official review of all these fines.
"If Mr Cummings' actions amounted to a 'reasonable excuse', including his test drive or day trip, how does this affect those convicted who might have put forward similar reasons?"
Ms Brimelow encouraged those who have been issued with fines to seek further guidance.
On Tuesday, Health Secretary Matt Hancock was asked by a member of the public at the daily coronavirus press briefing whether the government would review fines for people travelling for "childcare purposes".
Mr Hancock said it was "perfectly reasonable to take away that question" and he would look at it with his Treasury colleagues.
But government sources later clarified that there would be no review of fines issued under the coronavirus regulations.
Speaking to BBC Breakfast, Communities Secretary Robert Jenrick said police forces were asking their officers to "engage in the first instance, to explain and to resort to fines only when absolutely necessary".