Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$, and $L<\mathbb F_p^n$ a subspace of dimension $m:=n/(p-1)$. Do there exist vectors $l_1,\dotsc,l_n\in L$ such that the matrix with $l_1,\dotsc,l_n$ as its columns has a nonzero permanent? Clearly, the answer is negative if $L$ is contained in a coordinate hyperplane, or in a hyperplane like $\{(x_1,\dotsc,x_n)\in\mathbb F_p^n\colon x_1=\dotsb=x_p\}$; are there other obstructions of this sort?

Is it possible to classify those subspaces $L<\mathbb F_p^n$ for which any square matrix of order $n$ with all its column vectors in $L$ has a vanishing permanent?

Notice that if the permanent vanishes for $l_1,\dotsc,l_n$ being the elements of some particular basis of $L$, with each element repeated $p-1$ times, then in fact it vanishes for any $l_1,\dotsc, l_n\in L$.

Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$, and $L<\mathbb F_p^n$ a subspace of dimension $d=n/(p-1)$. Do there exist vectors $l_1,\dotsc,l_n\in L$ such that the matrix with $l_1,\dotsc,l_n$ as its columns has a nonzero permanent? Clearly, the answer is negative if $L$ is contained in a coordinate hyperplane, or in a linear subspace like $\{(x_1,\dotsc,x_n)\in\mathbb F_p^n\colon x_1=\dotsb=x_p\}$; are there other obstructions of this sort?

Is it possible to classify those subspaces $L<\mathbb F_p^n$ for which any square matrix of order $n$ with all its column vectors in $L$ has a vanishing permanent?

Notice that if the permanent vanishes for $l_1,\dotsc,l_n$ being the elements of some particular basis of $L$, with each element repeated $p-1$ times, then in fact it vanishes for any $l_1,\dotsc, l_n\in L$. (It is this property that depends critically on the assumption $d=n/(p-1)$.)

In the case where $d=1$, the requirement that $L$ is not contained in a coordinate hyperplane is easily seen to be also sufficient. The case $d=2$ does not look that easy to me.