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