= final
peremptory absolute, final and not entitled to delay or reconsideration. The term is applied to writs, juror challenges or a date set for hearing. https://dictionary.law.com/Default.aspx?selected=1500

Я вашу фразу понял так:

Адвокат подсудимого/ответчика отметил, что он намерен ходатайствовать о признании любого отложения судебных слушаний последним, и рассмотрение дела было в конечном итоге отложено в связи с необходимостью получения указаний директора государственного обвинения до 21 сентября (сего или следующего года), которое было назначено предельным/крайним сроком для стороны государственного обвинения.

Из следующего отрывка понятно, что в данном случае имеется в виду:

Addressing the court yesterday, Insp Tom Kennedy said the extensive file in relation to the case was with the Director of Public Prosecutions (DPP) to decide if further charges or amended charges should be brought against the five.
Insp Kennedy said that books of evidence in the case were being prepared.
However, a solicitor for three of the five, Mr Ted McCarthy, told the court all five had been in custody since January 23rd, and with four of the accused being young persons, it made it more urgent on the State to serve the books of evidence.
Judge Joseph Mangan said he would make the serving of the books of evidence on April 23rd peremptory against the State, which means if the State cannot produce the books of evidence on that date, the charges will be struck out.
Gardaí have the option of rearresting the accused and beginning the process once more. In response to making the adjournment peremptory against the State, Insp Kennedy said to Judge Mangan: "I would ask you not to do that."
However, Judge Mangan made the order and remanded the five in custody to reappear at Ennis District Court on April 16th for the matter to be mentioned and the books of evidence to be before the court on April 23rd. https://www.irishtimes.com/news/judge-sets-deadline-in-cases...