We study the representation theory of the infinite type A Hecke algebra over a non-archimedean field in the case where the parameter is a pseudo-uniformizer. Specifically, we consider a family of representations, called almost-symmetric, which satisfy additional topological and algebraic constraints. We construct a family of irreducible almost-symmetric representations indexed by integer partitions which arise as topological completions of specific direct limits of Hecke-Specht modules. Our main result is that every irreducible almost-symmetric representation contains one of these constructed irreducibles as a dense submodule. We give detailed analysis of these representations and construct functionals analogous to finite Hecke algebra traces.