Police have warned one person and two others are under investigation for naming Grace Millane's killer on social media during his High Court trial.
The 27-year-old man was convicted of murdering the British backpacker on Friday after a three-week trial in Auckland.
The man's name and any details that may identify him remain suppressed by the court. But some people have continued to ignore that court order, as well as warnings from police, by naming the killer online.
Ms Millane's final hours were carefully examined at her killer's High Court trial and a large media contingent reported on almost every detail - except for the man's name.
It was - and remains - suppressed by the court, but that hasn't stopped some international news sites and members of the public from naming him online.
This has been an ongoing problem since his arrest last December.
At the man's first court appearance, someone in the public gallery snapped a picture of him and posted it online, while others repeatedly shared his name on social media. The breaches prompted warnings from lawyers, judges, and police.
By May of this year, six months after Ms Millane's murder, one person had been warned and police inquiries into other incidents were ongoing.
The man's three-week High Court trial this month provoked another surge of people to again breach the suppression order on social media.
Justice Minister Andrew Little told Morning Report he expected New Zealanders who shared the name on social media would face legal action.
"I understand authorities will be looking at situations where that has happened in the last couple of days. Contempt of court is a serious matter, courts take it seriously when those cases are brought before them but ultimately that's a matter for the police."
Over the weekend, police said they were aware Ms Millane's killer had been named on social media in New Zealand and reminded the public that breaching court ordered suppressions was an offence.
Anyone who breaks the law in New Zealand could face a fine of up to $25,000 or six months in jail.
Police now say they have warned one person and are still investigating two others for naming the man online during the trial.
Anyone who posts his name online risks being charged, they said.
But criminal barrister Todd Simmonds said charges for breaching suppression orders were rarely laid.
"I think historically that really reflects the fact that court orders are treated seriously and they are certainly complied with as they should be."
Those who point others in the direction of his name online were not likely to be charged either, Mr Simmonds said.
"In general terms, it would require publication of name, address, occupation, or other information that would likely reveal the identity of the subject matter."