Structure of the Components of the Fixed Locus of a Self-Map of the Berkovich Line

We describe the local and global structure of the fixed locus for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. This includes a bound for the number of connected components that is sharp when the residue characteristic is large or zero. The case of small nonzero residue characteristic will be treated in a subsequent article.