Is a possessive noun a contraction?

I was told not to use contractions in an essay.

My classmate wrote "the argument of Emily" but I preferred "Emily's argument". He disagreed and claimed "Emily's" is a contraction.

"Emily's" can be a contraction – like when you're saying:

Emily's going with us tomorrow.

However, you've used a possessive, which is not the same thing as a contraction.

Remember, if you've used a contraction, you should be able to split the word back into two:

Emily is going with us tomorrow.

But you can't do that with "Emily's argument."

So, I could say:

Your friend's wrong.

or, I could say:

Your friend's argument is wrong.

but I'd only be using a contraction in the first case.

Well, if one wanted to be very picky, Emily's argument is a contraction of the Old English possessive form -es.

However, unlike the case of something like:

Emily's a nice girl

which would be a contraction of

Emily is a nice girl

the 's is now considered just a suffix and is not really a contraction of any modern English word.

Wikipedia reports:

In Old English, -es was the ending of the genitive singular of most strong declension nouns and the masculine and neuter genitive singular of strong adjectives. The ending -e was used for strong nouns with Germanic ō-stems, which constituted most of the feminine strong nouns, and for the feminine genitive singular form of strong adjectives. In Middle English the -es ending was generalised to the genitive of all strong declension nouns. By the sixteenth century, the remaining strong declension endings were generalised to all nouns. The spelling -es remained, but in many words the letter -e- no longer represented a sound. In those words, printers often copied the French practice of substituting an apostrophe for the letter e. In later use, -'s was used for all nouns where the /s/ sound was used for the possessive form, and the -e- was no longer omitted. Confusingly, the -'s form was also used for plural noun forms. These were derived from the strong declension -as ending in Old English. In Middle English, the spelling was changed to -es, reflecting a change in pronunciation, and extended to all cases of the plural, including the genitive. Later conventions removed the apostrophe from subjective and objective case forms and added it after the -s in possessive case forms.