An investigation by The Yorkshire Post also uncovered confusion among local authorities about just who is responsible for enforcing the legislation.
Critics say the single fine issued is proof that the ban was both unnecessary and unenforceable, but advocates counter that it is proof of pubic support and changing attitudes.
Simon Clark, director of smokers' group Forest, said: "The figures vindicate the argument that a ban on smoking in cars carrying children was unnecessary and would be impossible to enforce.
"The reality is that even before the ban very few adults were smoking in cars with children present. The overwhelming majority of smokers knew it was inconsiderate and had stopped doing it.”
The view that most families already avoided smoking in cars is shared by Public Health England, but it maintains the change in law was still needed.
Scott Crosby, who leads Public Health England's work on smoking in the region, stressed that tobacco smoke seriously harms young lungs.
“The law sent a clear message to people that it is not acceptable to harm children in that way,” he said.
“For health, the measure of success is the change to attitudes and behaviour, not the number of enforcement actions.”
Wakefield, East Riding and Scarborough councils all responded to Freedom of Information requests by saying that police were responsible for enforcement, while Harrogate said police held the "primary responsibility".
Yet the region's police forces say their officers cannot issue fines in relation to this piece of Department for Health legislation.
Instead, officers can give a verbal warning or guidance, notify the relevant local authority and even make a report to the courts if necessary.
South Yorkshire Police said verbal warnings would only be noted in an officer’s pocket book and not on a ticket, and neither Humberside nor North Yorkshire held data.
However, West Yorkshire Police did have two records on its system of verbal guidance issued in Leeds district - one instance in 2015/16 and another in 2016/17.
Public Health England pointed towards a YouGov survey which found more than 90 per cent of adults in our region supported the ban on smoking in cars carrying children.
It is this high support for the legislation that health charity ASH says is behind the very low number of fines issued.
Chief executive Deborah Arnott said: “Compliance with the legislation on smoking in cars with children is dependent on the level of public support not enforcement action. And people are complying with this popular law which protects children from the harm caused by secondhand smoke."
She argued that there is now growing support for a ban on smoking in all cars, quoting a survey which found 62 per cent of adults in England backed the idea in 2016 compared to 45 per cent in 2009.
“Prohibiting smoking completely would make the law simpler to enforce as well as protecting all car occupants from the harmful toxins in tobacco smoke,” she added.
But Mr Clark said: "The only impact the law has had is to stigmatise ordinary, decent people who don't need the government telling them how to behave in their own private space.
“Let this be a lesson to politicians and campaigners who want to extend the smoking ban to other areas. Allow smokers to use their common sense when lighting up and most will behave responsibly and with consideration for others.
"The last thing we need are more rules and regulations governing people's behaviour."