Abstract Let $${\mathcal {K}}$$ K be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $${\mathbb {R}}^n$$ R n , there is a well-studied notion of “ultrametric orthogonality” in $${\mathcal {K}}^n$$ K n . In this paper, motivated by a question of Erdős in the real case, given integers $$k \ge \ell \ge 2$$ k ≥ ℓ ≥ 2 , we investigate the maximum size of a subset $$S \subseteq {\mathcal {K}}^n {\setminus }\{\textbf{0}\}$$ S ⊆ K n \ { 0 } satisfying the following property: for any $$E \subseteq S$$ E ⊆ S of size k, there exists $$F \subseteq E$$ F ⊆ E of size $$\ell $$ ℓ such that any two distinct vectors in F are orthogonal. Other variants of this property are also studied.