I am interested in the following generalization of the first derivative test for real functions: Let $K$ be an ordered field and $p\in K[X]$ a polynomial. Consider an interval $I\subseteq K$ on which we have $p'\geq 0$ everywhere. Is $p$ necessarily monotone (i.e. non-decreasing) on $I$? If not, are there stronger conditions in the same direction that make this true (e.g. $p'>0$ or $p'\geq c>0$ everywhere)?
Some context and partial progress:
The statement for $K=\mathbb{R}$ follows directly from the mean-value theorem. For other ordered fields, there is no comparably general theorem though, as we may always find non-monotone functions that are locally monotone everywhere (since every other ordered field is disconnected in the order topology).
Nonetheless, the statement in question does hold for a lot of ordered fields, by a rather uninsightful argument: For polynomials of a fixed degree, the assertion can be expressed as a first-order statement in the language of ordered fields. Since all real closed fields are elementary equivalent, the statement holds for all of them (since it holds in $\mathbb{R}$). Furthermore, if the statement holds in $K$, it also holds in every dense subfield of it (since polynomial functions are continuous). As such, the answer is yes for every $K$ that sits dense in its real closure. (Note that this is not true for all ordered fields, e.g. $K=\mathbb{R}(T)$ with $T>\mathbb{R}$ does not have this property.)
I suspect that there is some rather more elementary argument, that settles it for all $K$, which eludes me so far. A (very much unexpected) counterexample would also be much appreciated.