Yea, floods due to malfunctioning sump pumps happen when you have long cold stretches. Same if you've got a condensate line for a dehumidifier in the basement (assuming it's damp enough to run in the winter).

If you don't see water lowering, and no water exiting, it's likely frozen. The motor cools itself by being submerged in water. But it's designed to run briefly. If it's running for hours, that water is going to heat up.

One option is to get a hose and connect it to the sump pump, then pump it somewhere else for the moment to keep things under control. Just make sure it's somewhere downhill so you're not just creating a loop. It's a shitty option, but a common one in weather like this. That hose might freeze too if it's kept full of water, but you could take it inside for a few hours so it can thaw, or just lift and let it drain before it freezes. This is actually a pretty common workaround. Some people do this every snowstorm. Depending on the home and land around it, alternatives are expensive. Some people also just have some extra pipe in the garage and unscrew the clamp on the exterior and route it through that makeshift pipe after a snowstorm. Then when the snow melts put it back to the original plumbing.j

Given things are pretty frozen, you could also get a bucket and just empty it a dozen times or so if a sink is nearby. Odds are the ground is relatively frozen, so after a dozen or so times there should slow down. In a pinch that's an option but IMHO too much work.

Hopefully later this week things will warm up during the day at least and give things a chance to thaw.

I'd look around and see if there's evidence the previous homeowner had an alternate way of getting rid of water. Some pipes stashed away in a corner or under a deck, some hose. It's not uncommon in extreme situations.