Police say multiple people may have been involved in the attack which killed an elderly woman and left her husband critically injured.
Freda Walker, 86, died of her injuries during an attack at her home while her 88-year-old husband Ken remains critically ill in hospital.
The incident happened in Langwith Junction, Shirebrook.
Officers described the killing as "an act of horrendous cowardice", having previously said that a "significant level of violence" had been used during the attack. It comes after Derbyshire Constabulary launched a murder investigation following the incident in Station Road at about 9am on Saturday, January 15.
In a press conference on Tuesday (January 18), Assistant Chief Constable Dave Kirby said police were following multiple lines of enquiry.
He issued an appeal to any members of the public who may have information to come forward.
He said someone responsible was out in the community and that police needed to "get hold of those people straight away".
He said Mr Walker remained in hospital in a critical condition.
"We're following a number of significant lines of enquiry; [there's] lots of work to do, a huge amount of dedicated resources in order to do that," he said.
"Ultimately, there's someone out there who is responsible for this tragedy - an act of horrendous cowardice and they're currently loose.
"They're currently out in the community. I'm not suggesting that people are now in danger of another attack or anything like that.
"But I am saying we need to get hold of those people straight away."
He said the couple were the 'backbone' of the local community.
"They've done so much work for the good of other people, that's what their lives were about," ACC Kirby said.
The couple had been married for 60 years and for all of that time lived in the same house.
Mrs Walker was a member of local clubs, including a knitting club.
Police also revealed that they are not formally linking the investigation to an incident that happened in the village of Trowell earlier this month, this case being investigated by Nottinghamshire Police.
A couple in their 80s were subjected to a vicious burglary in the village on January 6.
On Monday (January 18), Nottinghamshire Police said they had refused to rule out the two cases were linked.
Derbyshire police said that certain aspects of the two incidents were "similar" but that they were not formally linking the two incidents.
ACC Kirby added: "That's ongoing, it's being led by Notts police and that was an incident at Trowell.
"The two investigation teams and the two SIOs are working very closely to make sure that all the information is shared and opportunities are shared across the two.
"At the moment, though, we're not formally linking those incidents albeit that some of the ways that the crimes took place, the MO, are similar.
"Ken is still in a critical condition; still seriously ill in hospital."
Police were also seen continuing the investigation in Trowell on Tuesday morning.
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