Applying the Pumping Lemma to aspecific string
The empty string is a legitimate string, just like the empty set is a legitimate set. We say that a word $w \in \Sigma^*$ can be pumped with respect to parameter $p \geq 1$ and a language $L$ if there exists a decomposition $w = xyz$, where $x,y,z \in \Sigma^*$ satisfy the following three requirements:

$|xy| \leq p$.
$y \neq \epsilon$.
$xy^iz \in L$ for all $i \geq 0$.

A mathematical definition hides nothing. A word can be pumped if it satisfies the definition as stated. No more, no less.