Where can I find a list of countries that do not have psychiatric hospitals? I've searched on Google and Wikipedia but could not find any links.
It's not a sortable list, but the WHO keeps track of the mental health institutions per country and has a report on each country here: http://www.who.int/mental_health/evidence/atlas/profiles/en/
They also provide a shaded map in their atlas which I copied below. You'll have to look up the reports on the lightly shaded countries to check wether they have any mental healthcare facilities at all.