This paper presents the foundations of p-adic modelling in genomics. Considering nucleotides, codons, DNA and RNA sequences, amino acids and proteins as information systems, we have formulated the corresponding p-adic formalisms for their investigations. Each of these systems has its characteristic prime number used for construction of the related information space. Relevance of this approach is illustrated by some examples. In particular, it is shown that degeneration of the genetic code is a p-adic phenomenon. We have also put a forward a hypothesis on the evolution of the genetic code assuming that primitive code was based on single nucleotides and chronologically first four amino acids. This formalism of p-adic genomic information systems can be implemented in computer programs and applied to various concrete cases.