Mr Justice Cregan ordered they be jailed for two weeks each because of their behaviour at a hearing involving their son and brother, Enoch Burke, who is in prison for disobeying a court order to stay away from the school at which he used to teach.
Mr Justice Cregan said it was clear to him that members of the Burke family were impervious to reason and were motivated by a faith-driven ideology, refusing to accept the rule of law.
He said the rule of law applied to them and to every other citizen, whether they accepted it or not.
Martina and Ammi Burke have not yet been found by gardaí, who tried to execute the order at their home in Castlebar on Friday, March 6. In the intervening period, questions have been raised as to why it has taken so long to find them.
The Irish Independent understands that gardaí are aware that Martina Burke has left the jurisdiction “several times”.
It is believed she has not been seen in the company of her husband in Castlebar, Co Mayo, in the two weeks since gardaí tried to execute the warrant.
Ammi Burke has also not been seen, but gardaí believe she was at the family home for a period.
Last weekend, Martina and her son Josiah, along with another member of the Burke family, were removed from the foyer of Tandragee Free Presbyterian Church by PSNI officers after trying to confront ministers.
They attended the church on Friday and Saturday, but were denied an audience with the ministers they wished to speak to.
The congregation was gathered for the official opening of Whitefield College of the Bible, the church’s new centre for the training of ministers.
Recently, members of the Burke family have targeted DUP members and Christian churches across Northern Ireland, whom they accuse of not living by the direct teachings of the “sacred gospel”.
Josiah Burke posted a lengthy statement and a video of the incident in Tandragee to various social media channels.
In the video, Martina Burke spoke at length despite being wanted by gardaí south of the Border. She admonished some members of the Free Presbyterian Church for having an event where dancing occurred. She said the Bible forbids dancing.
One observer, who is acquainted with the Burkes, said he noted that Martina Burke was “uncharacteristically polite” to the PSNI in the videos posted by the Burkes last weekend.
“She normally berates and insults gardaí. I think it is interesting that she has modified her behaviour. I also think it is interesting she didn’t show up in court when the committal order was issued,” they said.
“Understandably, I think she does not want to go to prison.”
A source in Castlebar said gardaí are sensitive to the fact Martina Burke is of advancing years and that they have no wish to come into conflict with her or the wider family, which is likely to be filmed and broadcast online.
“If they could be tracked down with as little fuss as possible, it would be ideal,” the source said.
When they initially tried to execute the committal order, gardaí did not enter the family home.
They have the power to do so, but it is understood they did not wish to inflame the situation.
When the time of their arrest comes, Martina and Ammi Burke will be transported to the Dóchas Centre, the female prison at Mountjoy in Dublin, and an official custody handover will take place.
A valid committal order from the court must be lodged with the governor and then the Burkes will be formally processed as prisoners.
Martina and Ammi will be searched by an officer of their own gender. They will also be fingerprinted and photographed. Then they will be brought to their cells.
They will also be assessed by a nurse or doctor.
If either woman says they feel unsafe, they can ask to be placed on a “protection” wing where they are segregated from the general population.
A source told the Irish Independent there is a “strict” regime in place at the prison, as it is currently “extremely busy” and overcrowded.
A garda spokesperson said: “An Garda Síochána has received a court order relating to contempt of court by two individuals.
“An Garda Síochána does not comment on the detail of ongoing operations.”
‘If either says they feel unsafe, they can ask to be placed on a wing where they are segregated from the general population’