According to the U.S. Department of Justice, "The term service animal includes individually trained animals that do work or perform tasks for the benefit of individuals with disabilities, including psychiatric, cognitive, and mental disabilities." However, "[a]nimals whose sole function is to provide emotional support, comfort, therapy, companionship, therapeutic benefits, or to promote emotional well-being are not service animals."
So yes, there are service animals for people with mental disabilities but like all other types of service dog they must be trained to perform tasks in order to qualify as service animals.
Remember that just having a mental illness doesn't make a person disabled. An impairment such as mental illness must substantially limit a person's ability to function in order to be considered a disability.
According to Service Dog Central, "It is not enough to have a mental illness to qualify as a person with a disability under the ADA. According to the NIMH, 26.2% of adults in the U.S. suffer from a mental illness in any given year, but only 6% are severely mentally ill. So more than three quarters of those with a diagnosed mental illness are not disabled by that illness and would not qualify to use a service animal even if they would benefit from one."