Invalid assumption.
Whether it can or not depends on the environment.
Most ordinary freezers are designed to operate within a relatively luxurious temperature controlled in-door environment. Stick one in the sun on a Florida back porch in August and you may well discover it can no longer successfully "extract energy from the side that is below freezing". Same thing (only in reverse) can happen with a home heat pump in an Alaskan winter.
There is another important element here --- energy loss and insulation.
At a certain temperature (whatever it may be), the energy lost to the surroundings will start to exceed the capacity of the heat pump. At some point; speaking in practical terms, the heat pump becomes essentially worthless. You need either a bigger heat pump or you need more insulation to stop the loss --- both of which cost money and can't be obtained/installed immediately.
Now, the original claim of "heat pumps don't work below freezing" is trivially wrong (and so the question by lasson was justified), but the overall point is that heat pump efficiency gets worse the more you need it in the case of keeping your home warm.