# Is pumping lemma not applicable for every 'long enough' string in the language?

I recently learnt that a subset of a regular set may not be regular. This is causing me confusion as I imagined if a set is regular then every string longer than $$p$$ can be pumped in the language. So does a non-regular subset have strings only length $$? But if they have length $$, can't they be trivially pumped?

I assume that by now you already know that the language $$L_1$$ containing strings with equal numbers of 1 and 0 is not regular. Notice that this is a subset of the language $$L_2$$ that contains all binary strings, which is trivially regular. It is the property of having equal 0 and 1 that makes it impossible to have a DFA/NFA that will only accept strings from $$L_1$$.
So if you treat all of $$L_1$$ as being binary strings (as elements of $$L_2$$), then all of them can be pumped with respect to this property. But if you consider them as having equal number of 0 and 1, then that is the time where pumping lemma fails.