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Are there any psychological models which do not presuppose the existence of the mind?

In the same way that there are theories of physics which do not presuppose the existence of a luminiferous aether, is the sort of thing that I mean.

Can there be, or is it impossible by definition?

The denial of mind is called Eliminative Materialism or "Illusionism":

Eliminative materialism is the claim that people's common-sense
understanding of the mind (or folk psychology) is false and that
certain classes of mental states that most people believe in do not
exist. ... modern physicists are eliminativist about the existence of
luminiferous aether. Eliminative materialism is the relatively new
(1960s–1970s) idea that certain classes of mental entities that common
sense takes for granted, such as beliefs, desires, and the subjective
sensation of pain, do not exist.

Although not denying the mind, Methodological Behaviorism (or the Watson school), one of the more popular psychological approaches in the first half of the 20th century, rejects introspection as a useful methodology for studying it:

Methodological behaviorism: Watson's behaviorism states that only
public events (behaviors of an individual) can be objectively
observed, and that therefore private events (thoughts and feelings)
should be ignored.

The modern behavior analysis that emerged from this school of thought, and is still very active today, remains based on a logical behaviorism premise that the study of behavior should not reference the mind:

Logical behaviorism is a theory of mind that mental concepts can be
explained in terms of behavioral concepts.

Theory of mind doesn't really "presuppose" a mind. Theory of mind is just a name for a particular cognitive capacity: the ability to know that you have a mental state and to realize that others have a mental state, and that their mental state is not the same of yours. Calling this ability "theory of mind" might be a bit confusing because it sounds like it's a scientific theory, but it's not, it's just referring to an ability to theorize (make educated guesses) about the contents of other peoples' mental states. It could have been named something else (see Why is the theory of mind named as such? as well). Therefore, it doesn't really make any sense to think about "alternatives" to theory of mind, maybe just alternatives to the name of the ability.

Theory of mind is really important to human social interactions, and impairments in theory of mind are part of the spectrum of social difficulties in autism.

As one example of how theory of mind works day-to-day, very young children who have not yet developed theory of mind have difficulty understanding that knowledge they have is not shared by everyone. If they see you hide a candy under one of two cups, they would be confused why someone else who wasn't in the room to see where the candy was hidden doesn't know which cup the candy is under (Frith and Frith, 2005, tell a similar story to explain theory of mind).

The nature of the mental state is not actually a part of the definition of theory of mind. You might be thinking of philosophy of mind.

Frith, C., & Frith, U. (2005). Theory of mind. Current Biology, 15(17), R644-R645.

Yes, depending on what "mind" means.

A common folk perception of the "mind" is that there is some sort of operator within the brain, a unified conscious homunculus. Instead, cognitive neuroscience informs us that the mind is a network of faculties — one of which is consciousness. Most other faculties, such as memory, perception, use of language, etc., aren't problematic, but consciousness is.

If I may, I'll rephrase your question to "Are there significant theories that deny the existence of consciousness", the answer is still, intriguingly, yes.

One is the attention schema theory, developed by Michael S.A. Graziano. Very, very, simply, Graziano posits that what we think of as consciousness is nothing more that our ability to pay attention to what we are paying attention to.

That's inevitably a gross oversimplification. The Wikipedia article on AST provides more depth, as does Graziano's essay in the New York Times: Are We Really Conscious?. His 2013 book, Consciousness and the Social Brain, is an engrossing treatise on the theory.

If “the mind” in your question means a non-material entity that exists in an animal and that can sense signals from the environment (see, hear, smell, etc.), operate the signals, resulting in various mental activities (thinking, solving problem, planning, remembering, recalling, etc.), and send signals to its effectors (hand, legs, lips, etc.) to respond to the environment, then the answer is yes. There is a physical theory (with scientific evidence and verifiable predictions) that proves that there is no need to presuppose the existence of something like a luminiferous ether or any other exotic entity to explain the mind and its existence – the mind is just the information-processing processes of the brain. The theory that proves this is “The Basic Theory of the Mind”.

This theory also proves that qualia (i.e., what the color red is like, what the happiness is like, what the thinking is like, etc.) and consciousness (i.e., the awareness and experiences of what the color red is like, of what the happiness is like, of what the thinking is like, etc.) are also physical phenomena that need no exotic entities to account for – they are just signaling patterns and signaling states of neural processes.

Other interesting concepts of this theory are as follows:

…

6. 
   

   Because a neural process that performs a certain function without qualia occurring and a neural process that performs the same function with qualia occurring have different information in the processes, they require different signaling patterns. Therefore, they have different physical effects on other neural processes, at least from the different effects of different signaling patterns. Qualia thus have physical effects.

   
   
   

   In addition, because qualia induce the consciousness neural process to create conscious awareness and experiences of themselves and because the consciousness neural process is a physical process, qualia cause changes in a physical process and thus have physical effects.

   
   
   

   It can be proved similarly that consciousness (conscious awareness and experiences) has physical effects.

   
   



7. The fact that qualia and conscious awareness and experiences of them occur in only the final-stage sensory perception and the highest-level cognitive and executive neural processes, which are the latest-evolved neural processes, and never occur in more primitive neural processes, such as the brainstem, cerebellum, and basal ganglia, indicates that they are results of nervous system evolution.




8. The fact that it took more than two billion years after life had appeared on earth and hundreds of millions of years after the nervous system had developed before neural processes that were advanced enough to create qualia and consciousness could emerge into existence also means that qualia and consciousness came into existence because of no other cause than evolution.




9. The fact that qualia and consciousness still exist today indicates that they have been selected to remain in the evolutionary process. This means that they must have effects that help increase the chance of survival of animals that have them. Qualia and consciousness thus are the evolved functions to help increase the chance of survival of animals, including humans, that have them.

….

Thus, it concludes that we (our minds with qualia and consciousness) are evolved physical entities to help increase the chance of survival of ourselves and our species and that you (your mind with qualia and consciousness) are here to help increase the chance of your own survival and your species.

There are several other theories about consciousness and qualia but none directly answer your questions about the mind as this theory does. I list some of them here in case you are interested in reading more about this matter:

1. Block N. Comparing the major theories of consciousness. In: Gazzaniga MS, editor. The Cognitive Neurosciences (Chap 77). 4th ed., Cambridge, MA: MIT Press; 2009:1111–1122.




2. Kanai R, Tsuchiya N. Qualia. Current Biology. 2012 May;22(10):392–396.




3. Orpwood R. Neurobiological mechanisms underlying qualia. J Integr Neurosci. 2007 Dec;6(4):523-540.




4. Seager W. Theories of consciousness. An introduction and assessment. 2nd ed. New York, NY: Routledge; 2016. ISBN 978-0-415-83409-4.