It is consistent with ZF that there is no nonprincipal (i.e., singletons have measure 0) finitely additive probability measure at all. This appears in *Definability of measures and ultrafilters* by Pincus and Solovay [J. Symbolic Logic 42 (1977), no. 2, 179–190, doi: 10.2307/2272118, JSTOR], but the exact attribution is complex. From the intro to that paper:

Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo-Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF+ DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF+ DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a
different method. Our construction will be sketched in 4.1.