= Solution
\b[No.] For every $n\geq2$, take
$$
a_j=e_j\quad(1\leq j\leq n-1),\qquad
a_n=e_1+\cdots+e_{n-1}
$$
in $\mathbb R^n$. These form a <minimally dependent vector family>: the full family is dependent because $a_n-\sum_{j=1}^{n-1}a_j=0$.
If the omitted <vector> is $a_n$, the remaining coordinate <vectors> are independent. If the omitted <vector> is $e_k$, a relation among the retained <vectors> has an $e_k$ coordinate coming only from $a_n$, so its coefficient must be zero; the coefficients of all retained coordinate <vectors> must then be zero. Thus every $(n-1)$-element subfamily is <linearly independent>, as required for a counterexample. When $n=1$, the empty subfamily is independent while the one-vector family consisting of the <zero vector> is not, so the answer remains negative.
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