The motorway is currently shut between junction 34 and junction 35 northbound, near Sheffield and Rotherham.
A spokesman for Highways England said : "This incident is led by South Yorkshire and it is anticipated that the road will be closed for a number of hours while police investigation works take place."
A diversion is in place - follow the hollow triangle symbol. Motorists are advised to exit the M1 at J34 (South) and head onto the A6109, then the A629 up to the M1 northbound at J35.
UPDATE at 5.30pm: There are delays right down to junction 33 and motorists are advised to avoid the route all together. The motorway is still shut northbound.