The following problem bears some similarity to the Additive Basis Conjecture [ALM91,JLPT92] saying (in characteristic $3$) that there is an absolute constant $N$ such that for any positive integer $m$, any bases $B_1,\dotsc,B_N$ of ${\mathbb F}_3^m$, and any $g\in{\mathbb F}_3^m$, there exist subsets $S_i\subseteq B_i$ with $\sum_{i=1}^N \sum_{b\in S_i} b=g$.

Does there exist a positive integer $m$ such that, letting $q:=3^m$, there are bases $B_1,\dotsc,B_{100}$ of the field ${\mathbb F}_q$ (over $\mathbb F_3$) with the following property: for any choice of the subsets $S_i\subseteq B_i$, we have $\sum_{i=1}^{100}\sum_{b\in S_i}b^2\ne 0$, unless all subsets $S_i$ are empty?

Notice that, assuming the Additive Basis Conjecture is true with $N=99$, the squares $b^2$ cannot be replaced with the first powers. (I can also prove this unconditionally.) On the other hand, it would be fine with me to have the squares replaced with any powers $b^k$ where the exponents $k$ come from the set $K:=\{3^j+1\colon j\ge 0\}$:

Does there exist a positive integer $m$ such that, letting $q:=3^m$, there are bases $B_1,\dotsc,B_{100}$ of the field ${\mathbb F}_q$ (over $\mathbb F_3$) and exponents $k_1,\dotsc,k_{100}\in K$ with the following property: for any choice of the subsets $S_i\subseteq B_i$, we have $\sum_{i=1}^{100}\sum_{b\in S_i} b^{k_i}\ne 0$, unless all subsets $S_i$ are empty?