The short answer
You can, with limits. A mildly lowered car on proper winter tires handles plowed Winnipeg streets fine; the problems are unplowed side streets, windrows, frozen ruts and the salt that corrodes coilover threads and adjusters. Set a winter ride height if your suspension allows it, run narrower winter wheels, protect the coilovers, and if the car is very low, store it for the season.
Where lowered cars actually struggle
Main roads in Winnipeg are plowed and usually fine. The trouble spots are residential streets that stay rutted for weeks, the windrow the plow leaves across your driveway, icy parking lot ramps, and the frozen ridges at intersections. A car with a few centimetres less clearance beaches itself on those, and a front lip or splitter becomes a snow plow. The ground clearance you lost is the exact amount of snow that will stop you.
Salt and coilovers
The mechanical problem is corrosion. Salt and brine get into coilover threads, lock rings and damper adjusters, and by spring they do not move. Seized adjusters mean you cannot raise the car back up, cannot corner-balance it and may end up replacing the units. If you run coilovers through winter, clean and grease the threads before the snow, fit thread covers or boots, and wash the underbody regularly. Spring-and-strut setups on lowering springs have fewer exposed threads and tolerate winter better.
Winter tires on a lowered car
Wide summer wheels with low-profile tires are the wrong tool in snow. A narrower winter wheel with a taller sidewall, within what the fenders and brakes allow, gives better bite, more clearance and some protection for the wheel when you find a pothole under the slush. Check that the winter setup does not rub at full lock with the lower ride height, and recheck alignment, because a change in ride height changes camber and toe.
Raise it for winter, if you can
Height-adjustable suspension exists for this reason. Raising the car a couple of centimetres for winter costs a little in looks and almost nothing in handling at winter speeds. It needs an alignment afterwards and another in spring, which is also when you want the suspension inspected for pothole damage anyway. Garage 46 sets winter and summer heights and aligns to specifications that suit each.
When to store it instead
If the car is static-dropped very low, has an expensive aero kit, or is simply the car you care about most, storage is the honest answer. Rust, stone chips from winter grit, salt in every seam and one hidden pothole can undo a build. Storing it properly costs far less than fixing that, and it means a daily driver with real clearance for January.