In spite of the analogies between $\mathbb{Q}_p$ and $\mathbb{F}_p ((t))$ which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for $\mathbb{Q}_p$ to the case of $\mathbb{F}_p((t))$ does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on $\mathbb{F}_p((t))$. We formulate an elementary property expressing this action and show that it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic.