It is well-known that various profinite groups appearing in anabelian geometry — e.g., the absolute Galois groups of p-adic local fields or number fields — satisfy distinctive group-theoretic properties such as slimness [i.e., the property that every open subgroup is center-free] and strong indecomposability [i.e., the property that every open subgroup has no nontrivial product decomposition]. In the present paper, we consider another group-theoretic property on profinite groups, which we shall refer to as strong internal indecomposability. This is a stronger property than both slimness and strong indecomposability. In the present paper, we examine basic properties of strong internal indecomposability and prove that the absolute Galois groups of Henselian discrete valuation fields with positive characteristic residue fields or Hilbertian fields [which may be regarded as generalizations of p-adic local fields or number fields] satisfy strong internal indecomposability.