An interesting question. The thing is, yes it will freeze but then, ice also evaporates. When it does, it is called, sublimation. Of course, it doesn't sublimate at the same rate as if it was warm weather. Furthermore, the rate of sublimation will also depend on the relative humidity.
If a cubic meter of air can't sustain more than 13 grams of water at 15 degrees Celsius, it is down to 5 grams at zero degree, i.e. freezing point. But the relative humidity can be 100 percent, even if the temperature is below freezing point. This is when e.g. freezing fog occurs.
In Norway, where I live, we have now heat dryers in our homes but in the past, everyone was drying laundry outside, even during the winter. It takes longer time but when the water turns ice, it also falls down from the cloth when a breeze shakes it.
Incidentally, this is how ocean sailors wash their clothes when at sea for weeks or months. They tow it behind the boat for a day or so. The dirt is then removed by friction. Then they hoist the clothes to the top of the mast. While there, it dries and the salt is shaken from the cloth. At deck level it simply wouldn't dry because the air is saturated with humidity and the salt would attract it.
A dry trousers or jumper is a luxury for a single-handed ocean circumnavigating sailor!