Could faculty members earn a degree while being employed?
In France, it is possible, in particular in order to get the "Habilitation à diriger des recherches (HDR)", which is a degree one can get after a PhD, and that demonstrates the ability to fully supervise a PhD student. 
It is required in order to apply for a full professorship position. 

In order to get the HDR, the lecturer must write a thesis and defend it in front of a committee, and needs to enroll in a university (and therefore is a student), usually the one where she is working (but I don't think it is required). Note that for the HDR, there is no class or lectures to attend, only the thesis to write.