Are there any security risks in using Elliptic Curves defined over fields $\mathbf{F}_{p^n}$ where $n>1$
There are concerns on curves defined over an extension field. In particular for those of a small extension degree.

For a curve with a subgroup of prime order $r$, generally the best algorithm to solve the discrete logarithm problem has a complexity $O(r^{1/2})$. But there are attacks that use the structure of the extension field to get an algorithm with a smaller complexity.


Weil descent. The finite field $\mathbf F_{q^n}$ can be viewed as a $\mathbf F_q$ vectorial space of dimension $n$: a polynomial equation defined over $\mathbf F_{q^n}$ can be viewed as $n$ polynomial equations defined over $\mathbf F_q$.
It is possible to construct, under some conditions, a homomorphism from an elliptic curve $E$ defined over $\mathbf F_{q^n}$ to the jacobian a hyperellipti curve $H$ (the group associated to the curve, in the case of elliptic curve it is the curve itself) over the smaller field $\mathbf F_q$.
$$
\varphi : E(\mathbf F_{q^n}) \to \mathrm{Jac}(H)(\mathbf F_q)
$$
When this can be achieved, we can use better algorithms to solve the discrete logarithm on $\mathrm{Jac}(H)(\mathbf F_q)$.
There are some limitations: for most curves $E$, the hyperelliptic curve produced has a high genus that makes algorithms much slower than on the elliptic curve.
Algebraic approach. Index calculs method brought a better than square-root complexity for the discrete logarithm on the multiplicative group of a finite field. The idea is to decomposed random elements over a factor basis, and with enough relations and linear algebra, the discrete logarithme is solved. For elliptic curves over an extension field $\mathbf F_{q^n}$, the factor basis can be defined as:
$$
\mathcal F = \{ P \in E(\mathbf F_{q^n}) \mid x(P)\in \mathbf F_q\}.
$$
Suppose $n=2$ and you want to decompose a point $R=(x_R, y_R)$ into two points of $\mathcal F$. There exist a polynomial $S_3$ such that if $S_3(x_1, x_2, x_3) = 0$, then there exists $P_1, P_2, P_3$ with $P_1 + P_2 + P_3 = \infty$. It can be used to solve the system
$$
\begin{array}{l}
S_3(x_1,x_2,x_R) = 0 \\
x_1, x_2 \in \mathcal F
\end{array}
$$
And $S_3$ can be viewed as $2$ polynomial equations over $\mathbf F_q$ and since $x_1$ and $x_2$ are already in the subfield, we have a polynomial system (non-linear, unfortunately) of $2$ equations and $2$ variables that can be solved by computing a Groebner basis. This can be generalized for bigger $n$.
This approach has lead to an algorithm of heuristic complexity $\tilde{O}(q^{2-2/n})$ which is faster than the square-root algorithm when $n\geq 3$. One drawback is that there is a constant $n!$ on the complexity.


You can find a survey from 2015, Recent progress on the elliptic curve discrete logarithmproblem that talks about the above and more.
Attemps have been made to apply the algebraic approach for curves over a prime field $\mathbf F_p$ but have failed (a factor basis that can be translated algebraically is harder to find without having a polynomial of high degree, which makes is very difficult to solve non-linear polynomial system).

To prevent the above attacks, it is recommanded to have a finite field of prime degree which is the case for most of binary standardized curves ($\mathbf F_{2^n}$ with $n$ prime), but not all, or use a curve defined over a prime field.