Summary: We present a new non-Archimedean model of evolutionary dynamics, in which the genomes are represented by \(p\)-adic numbers. In this model the genomes have a variable length, not necessarily bounded, in contrast with the classical models where the length is fixed. The time evolution of the concentration of a given genome is controlled by a \(p\)-adic evolution equation. This equation depends on a fitness function \(f\) and on mutation measure \(Q\). By choosing a mutation measure of Gibbs type, and by using a \(p\)-adic version of the Maynard Smith ansatz, we show the existence of threshold function \(M_c(f,Q)\), such that the long term survival of a genome requires that its length grows faster than \(M_c(f,Q)\). This implies that Eigen’s paradox does not occur if the complexity of genomes grows at the right pace. About twenty years ago, Scheuring and Poole, Jeffares and Penny proposed a hypothesis to explain Eigen’s paradox. Our mathematical model shows that this biological hypothesis is feasible, but it requires \(p\)-adic analysis instead of real analysis. More exactly, the Darwin-Eigen cycle proposed by A. Poole et al. [BioEssays, 21, 880–889, (1999)] takes place if the length of the genomes exceeds \(M_c(f,Q)\).