It has been a long time since the world has experienced a pandemic with such a rapid devastating impact as the current COVID-19 pandemic. The causative agent, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is unusual in that it appears capable of infecting many different mammal species. As a significant proportion of people worldwide are infected with SARS-CoV-2 and may spread the infection unknowingly before symptoms occur or without any symptoms ever occurring, there is a non-negligible risk of humans spreading SARS-CoV-2 to wildlife, in particular to wild non-human mammals. Because of SARS-CoV-2's apparent evolutionary origins in bats and reports of humans transmitting the virus to pets and zoo animals, regulations for the prevention of human-to-animal transmission have so far focused mostly on these animal groups. We summarise recent studies and reports that show that a wide range of distantly related mammals are likely to be susceptible to SARS-CoV-2, and that susceptibility or resistance to the virus is, in general, not predictable, or only predictable to some extent, from phylogenetic proximity to known susceptible or resistant hosts. In the absence of solid evidence on the susceptibility and resistance to SARS-CoV-2 for each of the >6500 mammal species, we argue that sanitary precautions should be taken by humans interacting with any other mammal species in the wild. Preventing human-to-wildlife SARS-CoV-2 transmission is important to protect these animals (some of which are classed as threatened) from disease, but also to avoid establishment of novel SARS-CoV-2 reservoirs in wild mammals. The risk of repeated re-infection of humans from such a wildlife reservoir could severely hamper SARS-CoV-2 control efforts. Activities during which direct or indirect interaction with wild mammals may occur include wildlife research, conservation activities, forestry work, pest control, management of feral populations, ecological consultancy work, management of protected areas and natural environments, wildlife tourism and wildlife rehabilitation in animal shelters. During such activities, we recommend sanitary precautions, such as physical distancing, wearing face masks and gloves, and frequent decontamination, which are very similar to regulations currently imposed to prevent transmission among humans. We further recommend active surveillance of domestic and feral animals that could act as SARS-CoV-2 intermediate hosts between humans and wild mammals.