We know that from axiom of choice (or just BPIT) we can deduce ultrafilter lemma, which states that every filter can be extended to an ultrafilter. From this lemma we can derive the existence of at least 2 different ultrafilters (example: take an ultrafilter containing set of even numbers and ultrafilter containing set of odd numbers. They are distinct, as otherwise it would contain a set together with its complement).
I was wondering if it is possible (without assuming ultrafilter lemma) that there exist exactly one ultrafilter on $\omega$, or maybe that from one ultrafilter we can always construct another? My exact question is: Is it consistent with ZF that there exists exactly one ultrafilter on $\omega$?