Dynamic systems in non-Archimedean number fields (i. e. fields with non-Archimedean valuations) are studied. Results are obtained for the fields of $p$-adic numbers and complex $p$-adic numbers. Simple $p$-adic dynamic systems have a very rich structure–attractors, Siegel disks, cycles, and a new structure called a “fuzzy cycle”. The prime number $p$ plays the role of a parameter of the $p$-adic dynamic system. Changing $p$ radically changes the behavior of the system: attractors may become the centers of Siegel disks, and vice versa, and cycles of different lengths may appear or disappear.