ECDSA over $\mathbb{F}_{p^n}$ for $n>1$. How to calculate $r$ and $s$
However, $x_1$ is an element of $\mathbb{F}_{p^m}$, which for $m&gt;1$ it doesn't make sense to compute it's remainder over an integer. 


True; when they defined ECDSA, they did not consider curves over extension fields.  This means that ECDSA just isn't defined in that case.

Now, it wouldn't be unreasonable to extend ECDSA to cover that case; the requirement for the $kG$ to $r$ mapping is that it $r$ be computationally uncorrelated to $k$; you might (once you've defined the field representation) treat $x_1$ as an $m$ digit number in the base $p$, and then take that value modulo $n$ (which might be what you suggested...).   Of course, any such extension would need to be analyzed before usage...

Also, as for your objection "$r d_A$ is a curve point", no, it is not.  $d_A$ is the private key, which is an integer between 0 and $n-1$