Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$).
1) Let $f: K^n \to K$ be a function such that its differential $Df$ is constant $0$.
Is $f$ constant on some non-empty open set?
(Notice that it's easy to produce a function as above for $K = \mathbb Q$ such that $f$ is not locally constant at $0$).
2) Let $f: K^2 \to K$. Assume that the functions $\partial f / \partial x$ and $\partial f / \partial y$ exist and are continuous. Is $f$ $\mathcal C^1$?
