In a word, no.
The patronus must take the form of something that creates positive feelings (a dementor is by definition the embodiment of unhappy feelings, primarily fear) and the shape of an animal with whom they share an affinity.
Not only would someone who had an affinity for a dementor be unable to create a patronus, it's also fairly questionable whether a dementor qualifies as a 'animal':
"This ancient and mysterious charm conjures a magical guardian, a projection of all your most positive feelings.
The Patronus Charm is difficult, and many witches and wizards are unable to produce a full, corporeal Patronus, a guardian which generally takes the shape of the animal with whom they share the deepest affinity" - Wonderbook : Book of Spells and
"A Patronus is a kind of positive force, and for the wizard who can conjure one, it works something like a shield, with the Dementor feeding on it, rather than him.
In order for it to work, you need to think of a memory.
Not just any memory, a very happy memory, a very powerful memory… Allow it to fill you up...
lose yourself in it...
then speak the incantation " Expecto Patronum " HP:PoA (Film)
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This would be a bit ironic.
It would be hard, but I personally don't see a reason why it wouldn't be possible.
As @Richard said, a dementor is the embodiment of unhappiness, but there are some people who feel happy when they are depressed.
There are stranger ways people can feel/behave.
Another issue with a Patronus being a dementor is why the person creating this Patronus would need to create one, if he/she had an affinity with dementors.
In this case, I would imagine he/
she wouldn't have many happy memories to draw on, though, again, there are stranger things in this world than a person being happy and depressed at the same time.
Also, there are other creatures that get repelled by Patronuses, so even if the person didn't need to create one to battle dementors, he/she may need one for these creatures.
Also, there is the matter of creating one for fun/training, as Dumbledore's Army did in Order of the Phoenix .
Regarding a dementor being an animal, I'm not sure, but as they are (slightly) intelligent, I don't think they are "merely" animals,
in the same way humans aren't "merely" animals.
That being said, I don't believe J.K. Rowling ever said a Patronus has to be an animal.
I don't see why, for example, a Patronus couldn't be a human, or a giant, etc.
For narcissists, like Gilderoy Lockhart, I imagine they could have a Patronus in the form of themselves.
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Dementors aren't animals so the answer is probably no.
To clarify they are amortal non-beings.
They are not animals or humanoids but spirts that are manifestations of despair.
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In the book Patronus Charms are only Animals, And since Dementors are non-animals because of their sentience they would not be a shape found in a patronus.
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Reading the previous answers, it should be possible.
Although I believe that it would be rare to the point of being so rare to survive the killing curse.
There are several answers above that state
the patronus is the creature it is because it is something that the person has a AFFINITY with.
This does not mean that the wizard/witch has to have memories with the dementor, but rather has similar traits as a dementor.
If one was to decribe a dementor, soulless is a good one.
To have less soul someone would need to make a horcrux, but with the horcrux come with less morals for the original soul.
If one was able to create a dementor patronus that wizard/witch would most likely have a horcrux.
The part that might deny this idea is 'can someone with less morals create a patronus at all?'
I believe
so if they could before.
If the person did not know their happy memory, then created a horcrux, they most likely would never be able to create a patronus.
Let's say Ermac the Evil (the only other known person to create a horcrux) was able to create a patronus before making a horcrux, there is a chance it would be a dementor.
Earlier someone said the Death Eaters don't use Expecto Patronum because they don't need to, not that they can't.
While it is most probable that Voldemort doesn't have a happy enough memory, doesn't mean other wizards/witches with horcruxes can't.
Another that could shoot it down would be 'can a dark wizard or witch use a patronus?"
Again yes, because Severus Shape , a known dark wizard, can.
Even if he is on the light side he still has the taint of dark magic, hence dark wizard.
And yes I saw that POTTERMORE says a dark wizard will conjure maggots that eat him/her.
But PotterMore is written by fans that INTERPRET the movies, books, and interviews.
People mess up and they make mistakes.
Even when Snape repents and goes good he still has the taint of dark magic.
If Shape had no taint on him he most likely would not be able to kill Dumbledore with the killing curse later that year.
Also he cast the patronus after casting the killing curse at Dumbledore to guide Harry to the forest to get the sword of Gryffindor.
Snape has dark magic taint.
Snape can use patronus before killing Dumbledore.
Snape can use patronus after killing Dumbledore.
There is a possibility that a dark wizard with a horcrux can create a dementor patronus.
FIN.
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Given how a wizard in canon made a Patronus that was a giant, and then Dumbledore's own is a Phoenix
it's possible, just unlikely.
Now what I want to know is could a non-existent thing could be a patronus?
The rules just say it has to give a feeling of happiness, emphasized text not that it has to be a real or mythological animal.
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My question is simple. We know that a patronus can take the form of a magical animal (Dumbledore's phoenix patronus for example). So, can it take the form of a dementor? It would be a little bit ironic.  Perhaps there isn't any canon answer, but I'd like to know.
Can a patronus take the form of a dementor?
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<harry-potter><dementors><patronus-charm>
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The patronus must take the form of something that creates positive feeling or happiness so no, this would not be possible. It is possible though difficult to contemplate a patronus taking the form of a dementor. Moreover, Dementors aren’t animals and a Patronus generally needs to take the shape of an animal they have an affinity with which brings about happiness. 
It is possible though difficult to contemplate a patronus taking the form of a dementor. Dementors aren’t animals and a Patronus generally needs to take the shape of an animal they have an affinity with which brings about happiness. 