I was hoping someone more knowledgeable in this topic would respond, but that hasn't happened. So I will take a stab at this based on a class I took in undergrad a few years ago.

The earliest story that was written down is the Epic of Gilgamesh. It's a pretty cool story, obviously descendant from a long line of oral traditions. It just so happens that Gilgamesh is the earliest written work of fiction that survived, that doesn't mean it's the earliest work of fiction ever written down.

More generally in relation to writing, some of the earliest evidence for written, portable documents may be cylinder seals or stamps from the Near East. There is some evidence that these were used like royal rings in later periods, to sign other documents or as identification. Cylinder seals depict a variety of different scenes and designs and would have been unique enough to identify individuals or small groups of individuals. (The images are from the Boston Museum of Fine Art's online collection catalog.)

Cuneiform is one of the earliest written languages with an alphabet. My course focused more on the alphabet than other forms of writing (e.g. hieroglyphs). Cuneiform is about 5000-6000 years old. It is a type of writing that was impressed into clay tablets, totally portable. The types of documents written in cuneiform are mostly lists of things, goods for trade. As I mentioned, the Epic of Gilgamesh is the earliest written work of fiction, and it is dated much later than the earliest writing. I'm having trouble pinning down a specific date for Gilgamesh, but the earliest evidence is sometime between 4000 and 3000 years ago.