Minimum pumping length of (01)*

Michael Sipser offers the definition:

The pumping lemma says that every regular language has a pumping length p, such that every string in the language can be pumped if it has length p or more. If p is a pumping length for language A, so is any length p′ ≥ p. The minimum pumping length for A is the smallest p that is a pumping length for A.

Now, (01)* in set notation is {€, 01,0101,010101….} Taking minimum pumping length = 1, according to the definition, we have the statement if a string in the language has length 1 or more, it can be pumped.

This statement is true for all elements of the above mentioned set, so can the minimum pumping length be 1?

p.s. the minimum pumping length for (01)* has been asked here before but it doesn’t answer my doubt that since the condition holds for minimum pumping length = 1, why is it not the answer?