In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to me like there would be a total order. E.g. under the 3-adic norm I think $1\leq 4$ and $4\leq 1$ (since $3^0$ is the highest that divides both) yet $1\not = 4$ so it's not total.
When you talk about the "p-adic" integers as an ordered group, is it implied that you "mod out" everything which isn't a power of $p$ so that the order is total?