(This answers the previous version of the question.)

For example, you may choose large enough $N$, say $N=(m-1)n(n-1)/2+1$, $N$ linear functions $\ell_1,\dots,\ell_N$ in general position on $\mathbb{R}^m$ (any $m$ are linearly independent) and take the following map from $\mathbb{R}^m$ to $\mathbb{R}^{1+N(n-1)}$:
$$
x\rightarrow (1,\ell_1(x),(\ell_1(x))^2,\dots,(\ell_1(x))^{n-1},\ell_2(x),\dots,(\ell_N(x))^{n-1}).
$$
For any $n$ distinct points $x_1,\dots,x_n$ in $\mathbb{R}^m$ there exists index $i$ such that $\ell_i(x_1),\dots,\ell_i(x_n)$ are distinct reals. Indeed, for any pair $(x_p,x_q)$ there are at most $m-1$ indices $i$ for which $\ell_i(x_p)=\ell_i(x_q)$, totally at most $(m-1)n(n-1)/2$ not appropriate indices. Now linear independence of images follows from linear independence of $n$ Vandermonde vectors $(1,\ell_i(x_j),(\ell_i(x_j))^2,\dots,(\ell_i(x_j))^{n-1})$, $j=1,2,\dots,n$.