  We show that for every $1<n<\infty$, there exits a Banach space $X_n$
containing proximinal subspaces of codimension $n$ but no proximinal finite
codimensional subspaces of higher codimension. Moreover, the set of
norm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but no
subspace of higher dimension. This gives a $n$-by-$n$ version of the solutions
given by Read and Rmoutil to problems of Singer and Godefroy. We also study the
existence of strongly proximinal subspaces of finite codimension, showing that
for every $1<n<\infty$ and $1\leqslant k <n$, there is a Banach space $X_{n,k}$
containing proximinal subspaces of finite codimension up to $n$ but not higher,
and containing strongly proximinal subspaces of finite codimension up to $k$
but not higher. Finally, we deal with possible infinite-dimensional versions of
the previous results, showing that there are \emph{non-separable} Banach spaces
whose set of norm-attaining functionals contains infinite-dimensional separable
linear subspaces but no non-separable subspaces.
