Lincolnshire Police said Daniel Boulton, 29, was detained in the Hubbard's Hill area of Louth around midday and will be questioned in custody.
Officers had been hunting Boulton, who knew the victims, in connection with their deaths.
The force's head of crime, Detective Chief Superintendent Andy Cox, said: "This has been a tragic, extraordinary, and worrying set of circumstances in equal measure.
"A woman and child have lost their lives in the most devastating way and I know that their loved ones will be feeling the magnitude of that today and our thoughts go out to them.
"We have also seen communities in Louth, and indeed further afield, who will undoubtedly have been anxious about the possible danger in the town since last night.
"To all of you I want to say thank you. You have been listening to our safety advice and sharing our appeals, all of which has helped us to do our job to the best of our ability."
A force statement said: "We have located and arrested Daniel Boulton, who is wanted for questioning in connection with a murder in Louth last night and the injuring of a police officer in Hubbard's Hills earlier today.
"Boulton, 29, was detained in the Hubbard's Hill area around midday today. He is currently in police custody, where he will be held for questioning."
Armed officers and several police vehicles were involved in the arrest at a farm in Hallington, near Louth.
Two officers in high-vis escorted a man into a police van around the back of a farm building.
He was wearing no shoes or socks, ripped tracksuit bottoms, an Adidas T-shirt and a Nike jacket.
Officers have begun conducting door-to-door inquiries around the area and are searching fields behind the farm buildings.
A neighbour near the scene of where the arrest took place said a person being detained had attempted to get into another neighbour's car.
The witness, who asked not to be named, said: "He was hammering on the window of the car saying 'let me in, let me in'.
"(The other neighbour) is so shook up, she fortunately had the central locking on and managed to drive off before calling police.
"Police found him in one of the farm buildings but they are still looking for something."
Earlier police said an off-duty police officer was believed to have been injured by Boulton as they responded to an “incident” in Hubbard’s Hills.