I've come to know that you can't define an order relation over the field of p-adic numbers that is compatible with the addition and multiplication according to the ordered field axioms. I was wondering if there actually was a subset which contains an isomorphic copy of $\mathbb Q$ and can be totally ordered (compatibly with addition and multiplication).
[Edit from comment below] In particular, is there a way to totally order some subring $T$ of the $p$-adics, with $\mathbb Q \subset T$, so that the order respects addition and multiplication, and there is some $\omega \in T$ greater than all the rationals in the order?