The story of Amadis de Gaula may have first been told in the 13th or 14th centuries.
The earliest known printed edition was printed in 1508, revised by Garcia Rodriguez de Montalvo, who wrote the fourth volume. It was so popular that Montalvo and other writers wrote books V through XI published from 1510 to 1546, so this could count as an early novel series.
Amadis de Gaula was a medieval romance of chivalry, and hardly the first.
Earlier medieval romances include the Vulgate Cycle.
The Lancelot-Grail, also known as the Prose Lancelot, the Vulgate Cycle, or the Pseudo-Map Cycle, is a major source of Arthurian legend written in French. It is a series of five prose volumes that tell the story of the quest for the Holy Grail and the romance of Lancelot and Guinevere. The major parts are early 13th century, but scholarship has few definitive answers as to the authorship. An attribution to Walter Map is discounted, since he died too early to be the author.
The Vulgate Cycle adds an intriguing dimension to the King Arthur tradition, perpetuating Christian themes by expanding on tales of the Holy Grail and recounting the quests of the Grail knights. During this period, material takes on even more historical and religious overtones with tales that include and deal both in the death of Arthur and Merlin (drawing all the way back to Nennius's Historia Brittonum).
The Vulgate Cycle combines elements of the Old Testament with the birth of Merlin, whose magical origins are consistent with those told by Robert de Boron, as the son of a devil and a human mother who repents her sins and is baptized. Merlin is transformed into a prophet and given the ability of seeing future events by God.
The Vulgate Cycle was subject to a 13th-century revision in which much was left out and much added. The resulting text, referred to as the "Post-Vulgate Cycle", was an attempt to create greater unity in the material, and to de-emphasise the secular love affair between Lancelot and Guinevere. It omits almost all of the Vulgate's Lancelot Proper section, but includes characters and scenes from the Prose Tristan. This version of the cycle was one of the most important sources of Thomas Malory's Le Morte d'Arthur.
So it took me just a few minutes to look up and copy data about two medieval series that could be considered to be book series. Depending on how closely one considers them to fit the definition of a modern book series, they could be the first novel series ever, or possibly centuries later than the first ever novel series.