The application was rejected on Wednesday by Madam Justice Rafferty, one of the High Court judges who quashed the original decision.
In a brief announcement, she said: "This application for leave to appeal to the Supreme Court is dismissed."
Ball now has the opportunity to apply directly to the Supreme Court for leave.
Speaking in court after the decision, he said: "This is not over, we do not give up.
"We are pursuing her, absolutely."
Madam Justice Rafferty explained that "the problem of misrepresentation during a political campaign is not new" and that Parliament has enacted laws to deal with "some false campaign statements considered illegal practice ". ".
However, she added that this did not include misrepresentation of publicly available statistics, and concluded that the District Judge's decision would have "extended the scope" of the offense. misconduct in a public service.
She also stated that it seemed "that there would have been no complaint" if Johnson had used a figure of "350 million a gross week, or 250 million net a week".