I've searched all over the place for an explicit answer to this question, but I can't find anything. 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.