Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D : | x | < 1 . Let Mult ( A , ‖ ⋅ ‖ ) be the set of continuous multiplicative semi-norms of A, let Mult m ( A , ‖ ⋅ ‖ ) be the subset of the ϕ ∈ Mult ( A , ‖ ⋅ ‖ ) whose kernel is a maximal ideal and let Mult a ( A , ‖ ⋅ ‖ ) be the subset of the ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) whose kernel is of the form ( x − a ) A , a ∈ D (if ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) ∖ Mult a ( A , ‖ ⋅ ‖ ) , the kernel of ϕ is then of infinite codimension). The main problem we examine is whether Mult a ( A , ‖ ⋅ ‖ ) is dense inside Mult m ( A , ‖ ⋅ ‖ ) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult a ( A , ‖ ⋅ ‖ ) in the whole set Mult ( A , ‖ ⋅ ‖ ) : this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) is defined by an ultrafilter on D, ϕ lies in the closure of Mult a ( A , ‖ ⋅ ‖ ) . Particularly, we shaw that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) . Thus, if every maximal ideal is the kernel of a unique ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) , Mult a ( A , ‖ ⋅ ‖ ) is dense in Mult m ( A , ‖ ⋅ ‖ ) . And particularly, this is the case when K is strongly valued. In the general context, we find a subset of Mult m ( A , ‖ ⋅ ‖ ) ∖ Mult a ( A , ‖ ⋅ ‖ ) which is included in the closure of Mult a ( A , ‖ ⋅ ‖ ) . More generally, we show that if ψ ∈ Mult ( A , ‖ ⋅ ‖ ) does not define the Gauss norm on polynomials ( ‖ ⋅ ‖ ) , then it is characterized by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ does not lie in the closure of Mult a ( A , ‖ ⋅ ‖ ) , then its restriction to polynomials is the Gauss norm.