Can we specify the pumping length while applying pumping lemma
The pumping lemma states that if $L$ is regular, then $\exists p.\forall w.\exists x,y,z. w=xyz \wedge$ they can be "pumped"

Thus to show that $L$ is not regular, we must show $\neg(\exists p.\forall w.\exists x,y,z. w=xyz \wedge$ they can be "pumped"$)$
which is equivalent to $\forall p. \exists w. \forall x,y,z.w=xyx\wedge($ all other requirements given in the lemma$)\rightarrow\exists.n$ they cannot be "pumped" n times

This shows us its nessecary to never assume what p is.

Furthermore, it explains why we need to find only one word (which depends on p).