Uh, no! Span and basis are are completely different concepts, but they are related, of course: The span of a set of vectors (which, let it be noted, need not be linearly-independent) is a vector space (typically a subspace of the vector space under consideration), namely the set of all linear combinations of (finitely many) vectors from that set; whereas a basis of a vector space is a set of linearly-independent vectors, the span of which is the entire vector space under consideration.
uh, yes, Mr. fantastic is correct (not to mention fantastic!). A basis for a space is a set that both spans the space and is independent. Given any spanning set, if it not already independent, there will exist a subgroup that is and thus is a basis. That is why a basis is "a smallest spanning set". (I'm not sure I would have used the term 'dimension'.)
You appear ("typically a subspace of the vector space under consideration" for span and "the entire vector space under consideration") to be asserting that a subspace cannot have a "basis"- which is not true.
I am only asserting that a span is not a basis: a span is a subspace, which is never a set of linearly-independent vectors (which is required of a basis).
Of course, any vector space, in particular every subspace of a vector space (thus even span S), has a basis.
Please clearly distinguish my saying that "span S is not a basis" (which is true, I claim) from my saying "span S does not have a basis" (which is not true, and which is not what I have claimed).
Being and having are rather different concepts: you have a computer (I presume), but it does not follow from this that you are a computer.