Quick disclaimer: Take all information about COVID-19 cautiously. A lot of studies are case reports, preprints, and/or not peer-reviewed. That does not mean the information in these papers is wrong, but they could just depict an exemption or be presumptive to begin with.
Now that we got that sorted: Does the coronavirus (SARS-CoV-2) affect the brain? It can probably. But it is important that you continue reading. Let me start:
It has been recently reported that a subgroup of patients develop hyper-inflammation besides respiratory distress. Increasing evidence also indicates that SARS-CoV-2 might invade the central nervous system (Li et al., 2020). Indeed, a preprint study by Mao and Colleagues (2020) showed that 36.4% (in a sample size of 214 people) of SARS-CoV-2 people displayed neurological manifestations in form of i.a. headache, nausea, and confusion.
A woman in her 50s presented to the hospital with a three-day history of fever, cough, and altered mental status (Case report from Poyiadij et al., 2020). She tested positive for SARS-CoV-2, and MRI and CT lead to the diagnosis of acute necrotizing encephalopathy (ANE). In her case, i.a. lesions (= brain cell death) in the medial temporal lobe and thalamus (a relay station for sensory information coming from the body to other pars of the brain) were detected. ANE is a rare secondary disease characterized by brain damage following viral infections with fever, associated cytokine storms that lead to blood-brain-barrier breakdown. It is often characterized by symmetric lesions involving the thalamus, brain stem, white matter, and cerebellum. If left untreated, it can lead to disturbance of consciousness, seizures, neurological deficits, and death.
MRI of lesions (T2 FLAIR hyperintensity) in the medial temporal lobe (A) and thalamus (E). Image: Poyiadij et al., 2020, Radiological Society of North America.
A cytokine storm can occur following any infection (however, it is rare!). It is basically an overreaction of the immune system: In response to an infection, the cells of the immune system produce the messenger molecules called cytokines that help to coordinate the fight against the infection. Just like a fire alarm. When the fire is gone, the alarm turns off. In case of a cytokine storm though, this alarm stays on for too long (and possibly too loud), causing too many firefighters (= immune cells) to come, which potentially leads to tissue damage. Cytokine storms are associated with many infectious diseases like influenza, SARS, and MERS and NOT only SARS-CoV-2. The virus from the flu pandemic in 1918 for example was also known to cause deadly cytokine storms.
The woman is not the only case of viral encephalopathy. The coronavirus was found in the cerebrospinal fluid (CSF) of a 56 y/o woman in Beijing and a 74 y/o man (Filatov, Sharma, Hindi, et al., 2020). The latter presented to the hospital with altered mental state and SARS-CoV-2 in the CSF, however no brain damage could be detected at that point.
Going back a bit further, to another virus from the coronavirus family (SARS/SARS-CoV), a mouse study was able to show that SARS could enter the brain via the nose (Netland et al., 2008). As anosmia (= the loss of the sense of smell) has been found to be an early symptom of SARS-CoV-2, the idea of SARS-CoV-2 entering the brain via the nose is not too far-fetched (but has to be investigated further). As the nose is connected to the brain via the olfactory bulb, it could easily provide an entry to the brain.
So, in summary: Yes, there are hints for SARS-CoV-2 leading to neurological manifestations and cytokine storm related encephalopathy. Neurological manifestations are NOT uncommon in infectious diseases in general (so it is NOT EXCLUSIVE TO COVID-19!). Encephalopathy is a RARE complication of infectious diseases. It does happen in a small subgroup of COVID-19 patients.