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.
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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.