Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not satisfy the Archimedean property.
Non-archimedean norms on a field are in bijection with with valuations via $v(a) = -\log |a|$. This is used in the connection between tropical geometry and non-archimedian amoebas. See this paper: http://www.math.washington.edu/~lind/Papers/amoebas.pdf