The three synoptic Gospels (Matthew, Mark and Luke) fit the conditions listed in the question.
- They are works of fiction (more about this below) based on another work of fiction, in that at least one of them used at least one of the other gospels as a source, even though there exist various conflicting theories exist about which one is older. However, Mark is usually considered the oldest of the three (see next point) and was therefore probably a source for the two other synoptic gospels.
- Composed around the same time: Most scholars believe the Gospel of Matthew "was composed between AD 80 and 90, with a range of possibility between AD 70 to 110". Most scholars date the Gospel of Mark "to c. 66–74 AD, either shortly before or after the destruction of the Second Temple in 70 AD." "The most probable date for [the] composition [the Gospel of Luke] is around AD 80–110, and there is evidence that it was still being revised well into the 2nd century."
- "The derivative was not authorized by the original author." We have no evidence that this would have been the case, especially since each gospel contains a lot of content that cannot be found in the other ones.
Since I did not expand on "fiction" in the first list item above, the following is an enumeration of elements from the gospels that can only be fictional:
- The healing of the mother of Peter's wife (on of the miracles) in Matthew 8:14–15, Mark 1:29–31 and Luke 4:38–39.
- Exorcism in Matthew 8:16–17, Mark 1:32–34 and Luke 4:40–41.
- Cleansing a leper in Matthew 8:1–4, Mark 1:40–45 and Luke 5:12–16.
- Healing a paralytic at Capernaum in (Matthew 9:1–8, Mark 2:1–12 and Luke 5:17–26.
- Healing a man with a withered hand in Matthew 12:9-13, Mark 3:1-6 and Luke 6:6-11.
- Exorcising a blind and mute man in Matthew 12:22-32, Luke 11:14-23 and Mark 3:20-30.
- Calming a storm in Matthew 8:23–27, Mark 4:35–41 and Luke 8:22–25.
- (Walking on water in Matthew, Mark, and John but not in Luke.)
- Exorcism of the Gerasene demoniac in Mark 5:1–20, Matthew 8:28–34 and Luke 8:26–39.
- Raising of Jairus' daughter in Mark 5:21–43, Matthew 9:18–26 and Luke 8:40–56.
- Healing a bleeding woman in Matthew 9:20–22, Mark 5:25–34 and Luke 8:43–48.
- Feeding the multitude (with just few loaves and a few fish) in Matthew 14:13-21, Mark 6:31-44, Luke 9:12-17 and John 6:1-14.
- Transfiguration of Jesus in Matthew 17:1–8, Mark 9:2–8 and Luke 9:28–36.
- Exorcism of a boy possessed by a demon in Mark 9:17–29, Matthew 17:14–18 and Luke 9:37-49.
- Healing a blind beggar near Jericho in Mark 10:46–52, Matthew 20:29–34 and Luke 18:35–43.
- The resurrection.
These events cannot be explained based on current knowledge of physics, biology or other sciences and are therefore categorised as fictional here. They are also examples of plagiarism and would be considered copyright violations in the modern era.
The Acts of the Apostles, dated to around 80–90 AD, are a sequel of the same fictional story.
In addition to the official or canonical books of the New Testament, there are a number of other works, such as the infancy gospels and the non-canonical gospels. One such work is the Gospel of Judas, which Rowan Williams, Archbishop of Canterbury, denied the historical credibility of the canonical gospels in his Easter sermon of 2006:
So it was no huge surprise to see a fair bit of coverage given a couple of weeks ago to the discovery of a "Gospel of Judas", which was (naturally) going to shake the foundations of traditional belief by giving an alternative version of the story of the passion and resurrection.
Never mind that this is a demonstrably late text which simply parallels a large number of quite well-known works from the more eccentric fringes of the early century Church; this is a scoop, the real, 'now it can be told' version of the origins of Christian faith.
Response to comments regarding the fictionality of the gospels
Labelling the gospels as fiction does not imply that they are entirely fictional. They are regarded as fiction here because they contain elements that cannot be historical based on current scientific knowledge and because they borrow from certain pre-existing literary traditions. Two books that discuss the latter aspect are Lords of the Scrolls:Literary Traditions in the Bible and Gospels by Donald K. Sharpes (Peter Lang, 2005) and The Making of the New Testament Documents by Edward Earle Ellis (Brill, 2002).
Labelling the gospels as fictional in this way does not imply that all of the New Testament is fictional (various literary genres are represented in both the Old Testament and the New Testament, so no overgeneralisations should be made or implied). Whether a text is fictional or not is determined on a case-by-case basis.
Setting apart texts merely because they are "presented as a factual account" (see Pete's comment) implies that the fictionality of a text should be determined based exclusively on internal evidence and not on external evidence. This would make it impossible to distinguish most fiction from fact; it would also make it impossible to prove many liars wrong (unless they contradict themselves). Looking only at how a text presents itself is clearly a flawed method for whether a text is fiction or not.
Basing the validity of a statement on how it makes someone feel (see Laurel's comment: "Your answer makes me uncomfortable.") is a fallacy. On Stack Exchange, the value of an answer is based on the quality of the arguments and evidence it produces, not on how it makes someone feel. Closing or deleting an answer based on how it makes someone feel looks like a violation of the rules of Stack Exchange.
Below is the original version of this answer, posted before the question was reworded to exclude the Gilgamesh stories.
The oldest examples that meet the conditions listed in the question are probably the Sumerian stories around Gilgamesh (or Bilgamesh) and Enkidu from Ancient Mesopotamia. These date from the first half of the second millennium B.C., beating Avellaneda by 30 to 35 centuries:
- The Sumerian poem ‘Gilgamesh and the Land of the Living’, found at Nippur, of which 175 have been preserved.
- The very fragmentary ‘Death of Gilgamesh’.
From around the same period, we also have Old-Babylonian texts or text fragments, especially the Pennsylvania tablet and the Yale tablet. It has been argued that the Old Babylonian fragments were part of larger poetic narrative, whereas the Sumerian texts probably weren't. For more about this question, see Is there really a single "Old Babylonian version" of the Epic of Gilgamesh?
(Sumerian is believed to be a language isolate, whereas Akkadian, of which Babylonian was a variant, was a Semitic language.)
Source: APPENDIX: SOURCES, in 'The Epic Of Gilgamesh' on Erenow.net.