Reference Request: Independence of the ultrafilter lemma from ZF

In any of the formulations mentioned (so far) in the comments, the ultrafilter lemma is independent of ZF but weaker than AC. That the strongest form (all filters can be extended to ultrafilters) doesn't imply AC is a theorem of J.D. Halpern and A. Lévy ["The Boolean prime ideal theorem does not imply the axiom of choice" in Axiomatic Set Theory, Proc. Symp. Pure Math. XIII part 1, pp. 83-134]. That ZF doesn't prove even the weakest form (there exists a nonprincipal ultrafilter on some set) is a theorem of mine ["A model without ultrafilters," Bull. Acad. Polon. Sci. 25 (1977) pp. 329-331], building on S. Feferman's construction of a model with no non-principal ultrafilters on the set of natural numbers ["Some applications of the notions of forcing and generic sets," Fundamenta Mathematicae 55 (1965) pp. 325-345].

If you're just interested in context, rather than proofs, Eric Schechter's book Handbook of Analysis and Its Foundations talks about the relation between existence of ultrafilters, various weak notions of choice, and standard results in analysis.

The axiom "There exists some nonprincipalultrafilter." is discussed in Horst Herrlich, "The Axiom of Choice" as WUF(?) and in Rubin and Rubin "Equivalents of the Axiom of Choice, II." under [206]. The latter is kind of the standard ressource for these kinds of questions.

Here are papers providing proofs:

For 1:

MR0480028 (58 #227) Pincus, David ; Solovay, Robert M. Definability of measures and ultrafilters. J. Symbolic Logic 42 (1977), no. 2, 179--190. JSTOR, doi: 10.2307/2272118

For 2:

MR0480027 (58 #226) Pincus, David . Adding dependent choice to the prime ideal theorem. Logic Colloquium 76 (Oxford, 1976), pp. 547--565. Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977. doi: 10.1016/S0049-237X(09)70445-8