That's the best way to guarantee that your drain pipes freeze up!
We always make sure that we leave the plug in our wash basins with this sort of weather as we've had both of them freeze up in the past, particularly the one in the downstairs bathroom as it exits the wall about 1.5m from the downpipe.
The water flow is slow enough for it to freeze before is gets to the down pipe. More water flows over this thin ice layer and freezes so the layer of ice gets thicker - and continues to do so until the pipe is blocked. Come the morning and you've got a plug of ice five feet long to get rid of ...