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In set theory without the Axiom of Choice, we consider  Ingleton's axiom which is the counterpart in ultrametric analysis of theHahn-Banach axiom. We show that in $ZFA$, set theory without the Axiom ofChoice weakened to allow "atoms", Ingleton's axiom does not imply the Axiom ofChoice (this solves in $ZFA$ a question raised by van Rooij (1992).  We also prove that in $ZFA$, the "multiple Choice" axiom implies theKrein-Milman axiom. We deduce that, in $ZFA$, the conjunction of theHahn-Banach, Ingleton and Krein-Milman axioms does not imply the Axiom ofChoice.

Multiple Choices imply the Ingleton and Krein-Milman axioms