"Book" in this case being a bound collection of pages made from paper/papyrus/vellum/other materials.
Probably the earliest surviving codex fragment is Schøyen Collection MS 108, a group of three copper pages from a codex written in Greek-Phoenician script dating to ca. 800 BCE, found in Egypt at Fayum. Two of the leaves are now held at Oslo.
Codices were popular in the Roman world as notebooks: the Romans used wooden tablets bound together in a codex and covered in wax. It was possible to write quick notes in the wax using a stylus, and the wax could be wiped clean to use fresh.
For codices written on papyrus or parchment, we have testimony of their existence dating to the first century CE, but we don't start seeing surviving pieces until the second century. Here's an older post of mine on the transition that we see in the Greco-Roman world in the 2nd-4th centuries.
It's very difficult to date manuscripts of that period more precisely than to the correct century, but here are some likely candidates for the earliest ones (i.e. ones that are marked as "second century", but not "late second century" or "second/third century"):
The strong showing of Christian texts here reflects the fact that early hellenophone Christians strongly favoured the codex format, perhaps at least partly as a reaction against Jewish Christians and against Judaism generally.
Edit: Above I omit two other 2nd century Christian manuscripts: P.Oxy. 4404, which Wikipedia dates to ca. 150, because the Oxyrhynchus edition dates it to the late 2nd century (and so does Wikipedia, on a different page); and P.IFAO 237b partly for the same reason, partly because that one's a scroll, not a codex.