A good class of examples of this is given by Clifford algebras: If $V$ is a real vector space with endowed with a quadratic form $q:V\to\mathbb{R}$, the algebra $Cl(q)$ is the algebra generated by the elements of $V$ subject to the multiplication rule $x^2 = -q(x)$. If $M$ is a $Cl(q)$-module, say $M\simeq\mathbb{R}^m$, then we have an inclusion $V\hookrightarrow\mathrm{End}(M)$ and the characteristic polynomial of $x\in V\subseteq\mathrm{End}(M)$ is easily seen to be $(t^2+q(x))^{m/2}$, so we have
$$
\det(x) = q(x)^{m/2}
$$
for all $x\in V$.

For example, if $V$ is $\mathbb{R}^8$ with its standard Euclidean quadratic form $q$, then $Cl(q)$ is isomorphic to $\mathrm{End}_{\mathbb{R}}(\mathbb{R}^{16})$, so we can take $M=\mathbb{R}^{16}$ (and every $Cl(q)$-module is $\mathbb{R}^{16k}$ for some integer $k$). Thus, in this case, we have $\det(x) = p(x)^8$ where $p(x) = |x|^2$ for all $x\in V$.

In general, when $V\simeq\mathbb{R}^n$ and $q_n:V\to\mathbb{R}$ is nondegenerate, the dimension of a minimal nontrivial $Cl(q_n)$-module grows (roughly) exponentially with $n$, so the minimal $m$ grows exponentially with $n$. This shows that there are nontrivial 'irreducible' examples with $\det(x) = p(x)^k$ for $k$ arbitrarily large and that there is no bound on the possible dimension $n$ of the subspace $V\subset\mathrm{End}(M)$.

**Remark**: Given a linear subspace $V\subset\mathrm{End}(\mathbb{R}^{m})$ such that there exists a polynomial $p:V\to\mathbb{R}$ and an integer $k = m/\deg(p)>1$ such that $\det(x) = p(x)^k$, we say that the pair $(V,\mathbb{R}^m)$ is *irreducible* if there is no nontrivial subspace $M\subset\mathbb{R}^m$ such that $x(M)\subset M$ for all $x\in V$ and $\det(x_{|M}) = p(x)^j$ for all $x\in V$, where, necessarily, $j = (\dim M)/\deg(p)$.

The interesting problem for linear subspaces $V\subset\mathrm{End}(\mathbb{R}^m)$ on which the $\det$-function is a higher power of a polynomial on $V$ is to classify the irreducible ones of maximal dimension for a given $m$.