The project between junctions 28 and 31 had been due to be finished by this autumn, according to the Highways England website.
A spokesman for Highways England has now said the project should be completed by the end of December.
Smart motorways convert the hard shoulder to a running lane to reduce congestion.
Drivers have complained of long delays on the stretch of motorway since work started last summer.
The Highways England spokesman said: “We are improving the M1 by making it a smart motorway between junctions 28 and 31.
“We have a 50mph speed limit in place for the safety of road users and our work force. This is while we complete the project, which will relieve congestion and smooth the flow of the traffic, improving safety and journey times.
“By finishing in December – weather permitting – it will mean that by the New Year motorists will see a fully operational smart motorway.”
Highways England says it has put up signs informing drivers how long the project will take.