Large plumes of smoke were seen emanating from the store, part of York's Clifton Moor retail park, after the blaze broke out on Saturday afternoon.
North Yorkshire Police said that the fire started in B&M's storage yard, with fire crews calling out 10 pumps, two aerial ladder platforms and two water bowsers in an attempt to bring it under control.
Roads surrounding the premises, including part of the A1237 between Wigginton and Clifton Moor, had to be closed as thick smoke created difficult driving conditions.
"Major roads" around Clifton Moor were reopened on Saturday night, North Yorkshire Police confirmed.
A spokesman for North Yorkshire Fire and Rescue Service added that the situation is now "under control" and that fire investigation officers have been called to the scene.
Speaking on Saturday evening, a Yorkshire Ambulance Service spokeswoman said that, although staff had attended the scene has a precaution, no one has been admitted to hospital.
In a tweet, B&M Stores said: "All of our colleagues are safe and fire and emergency services responded as soon as possible."