The straight answer is no. The definition of a glacier requires the body to be of ice and snow, to move and deform under its own weight. If the body is pure ice the pressure needed to make ice deform corresponds to about 30-40 m of ice pressure.
The definition makes any body not consisting of ice and snow not a glacier. As an example, so-called "saltiers" are bodies of rock salt that move like glaciers. You find such features in the southern part of Iran. They would not qualify as glaciers from the first part of the definition.
The second part says that they must move. In mountainous terrain snow packs and also thin features of ice can move because they slide downhill. Because these are too thin there is no internal definition so the third part of the definition prevents such features from being glaciers.
Hence what remains are bodies of ice and snow that have sufficient thickness to induce deformation and hence they will be of a minimum thickness.
As a glaciologist, I do not find this definition very academic. With glaciers reducing in size around the globe many glacier will, in time, pass the limit and cease to be glaciers but knowing when this happens requires physical measurements which for practical reasons (and current technology) can only be made on a very small subset of the about 160 000 glaciers (not counting the ice sheets) on the earth. It is hence of little practical use when trying to figure out when particular glaciers cease to fit the definition.