Senior police figures and politicians have warned against speculation during the murder investigation into Ann Widdecombe’s death, after detectives said there was “nothing to suggest” political motivation after an intervention from Nigel Farage.

Devon and Cornwall police said on Sunday the killing investigation was not being treated as terrorism nor as politically motivated. Officers said they remained open-minded about the motive and urged the public not to speculate, warning it was both unhelpful to the investigation and distressing for Widdecombe’s family.

Widdecombe, a former Conservative minister and later Reform UK spokesperson, was found dead at her home in Haytor in Devon on Thursday having sustained serious injuries. Police said she may have been dead for more than 24 hours before her body was discovered.

Farage visited the area around Widdecombe’s Devon home shortly after news of her death. On Saturday, the Reform UK leader told journalists her death appeared to be “premeditated murder”. He also argued the case demonstrated that for “people now in public life, especially in politics, the world is very much more dangerous than it’s ever been, whatever the outcome of the motives of the killer”.

Police said a 28-year-old man arrested in Rotherham on suspicion of murder remained in custody. Officers said they were not looking for anyone else in connection with the investigation and there was no evidence to suggest any wider threat to the public.

Politicians from across the political spectrum have urged greater restraint, with one Labour minister saying that public statements “rarely help the police during an investigation”, while the former Conservative justice secretary David Gauke said people in public life “should know better than to speculate publicly”.

Sir Peter Fahy, the former chief constable of Greater Manchester police, said there had been a “very noticeable trend” for politicians, including government ministers, to comment on police incidents and murder investigations while they were still unfolding.

“It really does completely misunderstand and neglect the nature of a murder investigation,” Fahy told the Guardian. “Often at the start it is a bit of a voyage of discovery. You’ve got to keep a completely open mind. There may be people arrested who are found not to be involved.”

“We’re asking the police to do an impossible job here, where everything they do is being commented on and speculated on and often filmed as well,” Fahy added.

He warned that public interventions could create practical dangers for detectives, including witnesses being “contaminated”, potential suspects being alerted and evidence being destroyed.

“Things can be said and lodged in the public mind and then, when somebody is arrested and the case comes to court, even though the story may be very different, that doesn’t get the same sort of coverage.”

The former police chief said social media had fundamentally changed the environment in which criminal investigations take place. “I think the whole sub-judice rule is under a massive amount of pressure … the reality is, often in high-profile cases, so much is said and publicised beforehand … the whole murder investigation process, the criminal justice system, the court process was designed for another era, something closer to Midsomer Murders.”

One former Home Office minister said it was “wrong for senior political figures to engage in uninformed speculation during a live investigation”, adding: “This is distressing to victims’ families and can cause unfounded public alarm. It is of course reasonable to ask questions and raise issues, but we should always be guided by the facts.”

Sir Geoffrey Clifton-Brown, the chair of the Commons public accounts committee and a long-serving Conservative MP, said: “The police have a difficult enough job to do … having to cope with political figures guessing at the motivation and potential suspect could thwart a genuine arrest and charge. We have a convention in the House of Commons that we do not comment on cases which are sub-judice.

“We should widen this not to speculate via social or any media while the police are actively investigating a serious crime such as this.”

A Labour minister said: “We all want the police to have the space to do their job. MPs gobbing off rarely helps the police during an investigation.”

Another senior Tory MP said he thought Farage had “brought this into his narrative. He needs a narrative to keep himself in the media for the next four weeks to talk about anything that isn’t [questions over the £5m gift].

“He’s leveraging this to say we all need security because this is ‘premeditated’. There’s no evidence to suggest that it’s a politically motivated murder at all. Until the police conclude their investigation, the speculation is just harmful.”

Speaking at a press conference on Sunday morning, the assistant chief constable of Devon and Cornwall police, Matt Longman, said: “At this point, there is still no information to suggest that this is a terrorism-related incident and at this point we are not looking for anyone else in connection with this murder.

“At this stage, there is nothing to suggest that it was politically motivated.”

Longman added: “We are aware of online and public speculation, particularly with regards to motive. Again, I urge people not to share or engage with that speculation. It’s unhelpful.

“It doesn’t aid our investigation, and particularly, it’s distressing to the family and friends of Ms Widdecombe.”

Fahy warned that Britain should protect itself against greater political control of policing, saying reforms must not erode operational independence.

“There is a real risk that our police are going to come under greater and greater political control, and that should be a real worry for everybody, whatever party … we could end up with something a little bit close to what ICE are doing in America.”

Fahy said politicians should resist the temptation to post on social media before investigators have established the facts and warned against people turning major police operations into political issues “for their own political ends”.

Reform UK has been approached for comment.