Suppose to have three different matrices $$ \underbrace{A}_{P\times J_a}, \;\; \underbrace{B}_{P\times J_b},\;\;\; \mbox{ and } \underbrace{C}_{P\times J_c}$$ with full column rank, since $P>J_a$, $P>J_b$ and $P>J_c$ we have $rk(A) = J_a$, $rk(B) =J_b$ and $rk(C) =J_c$.
If we assume that the intersection of the span of the three matrices is empty, $span (A) \cap span(B) \cap span(C) = \emptyset$, is that true that the column vectors of the three matrices are linearly independent? Can you prove that?
Consider the next counterexample:
$A=\begin{pmatrix}1 \\ 0 \\ 0 \\0 \\ 0\end{pmatrix}$ $B=\begin{pmatrix}1 & 0 \\ 0 & 1 \\ 0 & 0 \\0 & 0 \\ 0 & 0\end{pmatrix}$ $C=\begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$
In this case the intersection between the column spaces generated by the three matrices is the empty set, that is $span(A)\cap span(B)\cap span(C)=\emptyset$. However, the first columns of $A$ and $B$ are not linearly independent.