Is there a language for which any string (that works) can be shown to not be pumped using the Pumping Lemma? Of course, whenever the Pumping Lemma for Regular Languages (or CFLs) is applied to a language $L$, only a single string $w \in L$ and $|w| \ge p$ needs to be picked (to eventually show that $w$, when pumped, leaves the language).

Is there a non-regular (or non-context-free) language $L$ for which every string $w \in L$ having $|w| \ge p$ can be used to derive a contradiction?
I'd say the language $L = $ { $0^n1^n | n &gt; 0$ } fits the bill for the Pumping Lemma for Regular Languages.  Any substring to be pumped would consist either of all the same symbol, in which case pumping up produces a string with an unequal number of $0$'s and $1$'s, or it contains instances of both symbols, in which case pumping up produces a string with a $1$ followed by a $0$; in neither case is the pumped-up string in $L$.