Is it known to be consistent with ZF that there is no non-principal ultrafilter on any infinite set? (Feel free to use your favorite interpretation of "infinite" in this context. If infinite just meant infinite ordinals, that would be fine, too. You may use all kinds of large cardinals.)
Yes, it is consistent with ZF that every ultrafilter is principal. This is a result of Andreas Blass, A model without ultrafilters, Bull. Acad. Polon. Sci. 25 (1977), 329–331.