Too long for a comment:

So you have $$M=\sum_{k=1}^{n-1}\binom{2^n}{k}\leq \binom{2^n}{n}$$ (for $n$ large enough) such possible (nonempty) sets $S$ and there are $$ F= \frac{ (q^m-1)(q^m-q) \cdots (q^m-q^{k-1})}{(q^n-1)(q^n-q) \cdots (q^n-q^{k-1}) } $$ distinct $n-$dimensional subspaces in the target space. Thus by a greedy method it looks like if $m$ is large enough for $M\geq F,$ to hold such a mapping can be found.

Simon McNicol et al, in Traitor tracing against powerful attacks, IEEE ISIT Proceedings 2005 (sorry can't find a free link yet) have defined $\delta-$nonlinear codes, as a code where for any collection of $\leq \delta$ codewords, the sum is not a codeword.

They take generalized Reed Solomon codes and use a concatenated construction together with a permutation of the codewords of the GRS.

The GRS code has alphabet $\mathbb{F}_{2^n}$ a permutation polynomial in $\mathbb{F}_{2^n}[x]$ is specified, and if the following conditions hold, there exists a code with the property you want. This code is over $\mathbb{F}_{2^n}$ so you'd need to represent codewords as $n-$ vectors which will multiply codeword length by $n$.

Theorem: If $2^n>r(s+1)-1,$ and $$N>\binom{\delta+1}{2} s,$$ then a $\delta-$nonlinear code derived from a GRS code exists. Here $r$ is the degree of the permutation polynomial used in the construction.

Conversion to binary means that the blocklength (your $m$) of the code is actually $nN.$

It would be interesting to look at randomized constructions which wouldn't have the structure in their construction (they wanted the distance distribution of the GRS code to be preserved) which would probably be more efficient.

I will give more details later when I have more time.