The PSNI has launched an investigation after images emerged of a procession at the funeral of former Sinn Fein councillor Francie McNally.
A senior police officer said he was disappointed advice given to family and friends of the late Mr McNally on social distancing was not followed.
Mr McNally died at Causeway Coast Hospital on Monday, aged in his early 60s. He was buried at St Patrick's Church in Ballinderry on Wednesday, April 8.
Pictures on social media showed a large crowd of people gathered for the procession. They lined up behind a horse drawn carriage.
Mid Ulster District Commander Superintendent Mike Baird said they were aware of plans for the funeral and had "engaged" with the family of the deceased, and with the local priest, "to stress the public health advice and risks around Covid-19 and the requirement for social distancing to be adhered to for those family members attending".
"The family had assured us that only family members would be in attendance, and that local people may pay their respects as the funeral cortege passed, by coming out into their gardens or the front of their homes," he added.
"It is very disappointing to see some people blatantly ignored health advice and breached current legislation by attending the funeral and, in doing so, they not only put themselves at risk but also put at risk close family members of the deceased and those officiating at the funeral."
Superintendent Baird said they were aware of social media commentary and images circulating online.
"An investigation is underway, evidence is being gathered and a file is being prepared for submission to the Public Prosecution Service for any breaches of Health Protection (Coronavirus Restrictions) Regulations NI 2020.
“I would also appeal to anyone who has any imagery or footage, or are aware of those who contravened the regulations and put other lives at risk to call us on 101."
DUP MLA Keith Buchanan said serious questions have been raised about adherence to regulations on social distancing.
"This was not even a spontaneous gathering of family or friends. There was an organised party of men accompanying the coffin alongside a relatively large gathering of other people in attendance," he said.
Mr McNally was a Sinn Fein councillor between 1985 and 1989. Two of his brothers were killed during the Troubles. Phelim McNally (28) was killed in a gun attack by loyalists at Francie McNally's home in 1988, while another brother Lawrence (39), an IRA member, was shot dead in Coagh in an SAS ambush.
In a statement, a Sinn Fein spokesperson added: "While we understand this unprecedented public health emergency has created enormous difficulties for families who have lost loved ones, particularly around the guidelines for funeral services, everyone should follow the public health guidelines on gatherings and physical distancing."
Archbishop of Armagh and Primate of All Ireland Eamon Martin said it is not the church's responsibility to police funerals.
After speaking to the parish priest Father Donnelly, he said the family of the late Mr McNally and the funeral directors were all in agreement the burial was being held for immediate family members only.
Father Donnelly confirmed to the Belfast Telegraph that 10 family members were present at the burial in the graveyard.
Archbishop Martin told BBC Talkback: "If a whole group of other people arrive on the scene, I don't think the priest who is in the middle of doing the funeral ceremony or the family who are bereaved, I don't think they can police these situations.
"All we can do is appeal to everyone to please respect the messages that are going out from our health authorities, our leaders in government to say listen, we are respecting these things to save lives. Some people just don't abide by restrictions."
The Archbishop said he believed the community had a role to play in restricting large gatherings at funerals, rather than the police.
Mr Buchanan said the grieving process has become more difficult for families due restrictions around coronavirus, but said the first priority must be to protect life.
He added: "The grief of all families who have lost a loved one at the current time is exacerbated by the stringent measures which have been put in place. However, those regulations apply to everyone regardless of status.
"As we approach this Easter weekend there is a particular concern that people may relax their guard in terms of social distancing. However the threat from this terrible disease means it cannot be business as usual.
"Unfortunately that must apply to all aspects of life, including funerals."