Can I use fat-free milk in béchamel?
Yes. The important fat is the one the starch first goes into. The higher temperature reached by fats (as compared to water or milk) helps cook the flour, creating flavors.  The liquid added later dilutes the mixture into a sauce. The liquid's fats are not as crucial for the sauce's thickness as the liquid's proteins.

I think of the cooked butter-flour mixture  as a thickner that can be added to milk, broths, and even a saucy dish.  The roux thickens way better than just a slurry of flour and a cold liquid.

Jack Lang has written a long post on Béchamel and Hollandaise and the science is summarized in chapter 4 of Vaclavik and Christian Essentials of Food Science