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.



Subspaces of vanishing permanent
Let $v\_1,\dots, v\_d$ be a basis for $L$. Then the permanent of the matrix obtained from $p-1$ repetitions each of $v\_1,\dots, v\_d$ is a polynomial function in the entries of $v\_1,\dots, v\_d$. Keeping all the entries but the last one in each $v\_i$ constant, we get a linear function. Since it is linear, among all $p^{nd}$ possible tuples of vectors, the number that satisfy it is at least $p^{nd-1}$.


Not all $d$-tuples of vectors in $\mathbb F\_p^n$ form a basis, but $1 - O( p^{d-n-1})$ do.


So the fraction of subspaces $L$ where this determinant vanishes is at least $$ \frac{1}{p} - O( p^{d-n-1}) $$


On the other hand, the fraction that satisfy one of your conditions is smaller than that for $d>1$ - just $\frac{ d (p-1)}{p^d}$ satisfy the coordinate condition and $\frac{ \binom{d (p-1) }{ p } } { p^{ (p-1) d }}$ satisfy the $p$-fold repetition condition.


So there certainly are other obstructions. I do not expect it is possible to classify them.


