In this short note we prove the result stated in the title; that is, for
every $p>0$ there exists an infinite dimensional closed linear subspace of
$L_{p}[0,1]$ every nonzero element of which does not belong to
$\bigcup\limits_{q>p} L_{q}[0,1]$. This answers in the positive a question
raised in 2010 by R. M. Aron on the spaceability of the above sets (for both,
the Banach and quasi-Banach cases). We also complete some recent results from
\cite{BDFP} for subsets of sequence spaces.