Apparently, there are 27 known copies of the first edition (1605). But Cervantes' hand-written original is not extant, correct?
In 2016, on the occasion of the 400th anniversary of Cervantes' death, Taberna Libraria published a facsimile edition of all of the author's manuscripts. This edition contains only 12 manuscripts. For a description, see the PDF file Autógrafos de Miguel de Cervantes Saavedra (in Spanish). The book was published in a limited edition of 1616 copies. King Felipe VI of Spain gifted one copy to Pope Francis.
The book was presented in the building of the Real Academia Española (RAE) in January 2016. It contains facsimiles of twelve manuscripts dated between 1582 and 1604, mostly letters and other documents. (The Quijote was published in 1605 but printing began in 1604.) It does not contain the manuscript of Don Quijote de la Mancha.
In 2007, Fracisco Rico, who edited the works of Cervantes, said that he would prefer not to see the manuscript of Don Quijote de la Mancha because it must have been a disaster from a modern point of view, for example because there was no fixed orthography at the time and because Cervantes wrote it with interruptions, in shops, in inns, while on the road, etc. without proper punctuation (at least, based on the manuscripts that have survived). (I assume some irony on Rico's part; in spite of the editorial difficulties, I assume he would still be curious.) He also says that a manuscript might exist somewhere.
Miguel de Cervantes I Saavedra was just a figurehead, who had to write down what Thomas Shelton dictated him, while being in the Netherlands. Shelton was the translator of ‘The History of the valorous and wittie Knight-Errant Don-Quixote of the Mancha’. Shelton received the chapters by the English Embassy in Brussels. Cervantes was also in the Netherlands between 1601 and 1604. By mid-1604 Miguel was back in Spain and offered the manuscript to the publisher. This precious book is the only surviving copy in the Biblioteca Nacional in Madrid (Cerv.118). (Tesoros de España, Ten Centuries of Spanish Books, lists the details of the first edition in Spanish. (24 p.293)) The permission to print the manuscript of DQ has the date 26 September 1604.
You can already read on the first page that
the first edition of the Quixote was carelessly done. Typographic, punctuation and foliation errors abound; accents are missing, and what is worse, parts of the original text, such as the passage in which Sancho Pança’s donkey is stolen by Ginés de Pasamonte, were not included... and (they) tried to withdraw the largest possible number of these books from circulation.
The first edition was published at the beginning of January, 1605 and delivered to ‘the Hermandad de Impresores de Madrid’ in the middle of May. The book came out without permission from the vicar of Madrid and without religious censorship or approval by the Royal Council. Juan Gallo de Andrada talks on 20 December 1604 about a book entitled: El ingenioso Hidalgo de la Mancha.. (no mention of the name Don Quixote) and the licentiate Murcia de la Llana stated in the fe de erratas that ‘this book has nothing that does not correspond to its original,’ sets the selling price at 290 and a half maravedíes.
Permission had already been given on 1 December 1604 from the Colegio de la Madre de Dios de los Teólogos de la Universidad de Alcalá, but no one else. Three times a confirmation is given and three times with the same wrong title.