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In [H-Y83], Herman and Yoccoz prove that for any given locally analytic (at$z=0$) power series $f(z)=z(\lambda +\sum_{i=1}^\infty a_iz^i)$ over a completenon-Archimedean field of characteristic $0$ if $|\lambda|=1$ and $\lambda$ isnot a root of unity, then $f$ is locally linearizable at $z=0$. They ask thesame question for power series over fields of positive characteristic. In thispaper, we prove that, on opposite, most such power series in this case are morelikely to be non-linearizable. More precisely, given a complete non-Archimedeanfield $\mathcal K$ of positive characteristic and a power series$f(z)=z(\lambda +\sum_{i=1}^\infty a_iz^i) \in \mathcal K[\![z]\!]$ with$\lambda$ not a root of unity and $|1-\lambda|<1$, we prove a sufficientcondition (Criterion~\star) for $f$ to be non-linearizable. This phenomenon ofprevalence for power series over fields of positive characteristic beingnon-linearizable was initially conjectured in [Her87, p 147] by Herman, andformulated into a concrete question by Lindahl as [Lin04, Conjecture 2.2]. Asapplications of our criterion, we prove the non-linearizability of threefamilies of polynomials.

Non-Linearizability of power series over complete non-Archimedean fields of positive characteristic