 Integer criterion for a non-Archimedean absolute value

An absolute value on a field is non-Archimedean if and only if $|n\cdot 1|\le 1$ for every positive integer $n$. Necessity follows from the ultrametric inequality. Conversely the binomial theorem gives $|x+y{|}^m\le (m+1)\max (|x|,|y|{)}^m$, since every binomial coefficient has absolute value at most one. Take $m$th roots and let $m\to \infty$. In particular an extension of an absolute value that is non-Archimedean stays non-Archimedean.