My question is rather simple. Do there exist a formally real field that admits a unique ordering (so sums of squares are the positive elements) and such that this ordering is not archimedean?

Oh, I have forgotten to add that the field I am looking for cannot be euclidean (in particular it cannot be real closed)!

The question I have in mind is the following. I proved many years ago that if K is a field that admits a unique ordering and this ordering is archimedean, then every automorphism f of K(x), where x is an indeterminate, maps K onto K. I wonder if archimedeanity can be dropped in this statement, and I realize that I know very few fields K whose unique ordering is not archimedean; essentially, real closed fields that cannot be embedded into the reals. And for such fields I know that f(K) = K!!

Puiseux series.) $\endgroup$ – Peter Heinig Mar 20 '18 at 10:24