Ingleton's theorem, a p-adic analog of Hahn-Banach's theorem, states that given a spherically complete ultrametric valued field K, and an ultrametric semi-norm p on a K-vector space E, any linear form f defined on a vector subspace V of E such that |f|≤ p can be extended into a linear form f̃ which is defined on E and such that |f̃| ≤ p. In his paper "The Axiom of Choice in p-adic functional analysis" (Lecture Notes in Pure and Appl. Math., 137, Dekker, New York, 1992), A.C.M. van Rooij studied Ingleton's statement in set-theory without the Axiom of Choice and asked several questions. We propose some additional results to this topic.