The COVID-19 pandemic generated high mortality in various countries, which may have had an impact on the mental health of young people. The objective of the study was to evaluate whether the death of a family member or close friend due to COVID-19 generated a higher prevalence of depression, anxiety, or moderate/severe stress in university health sciences students in Latin America. This is an analytical cross-sectional study, with secondary data; depression, anxiety, and stress were measured with a validated survey. In addition, data were obtained on the deaths by COVID-19 of family members or close friends, illness and other socio-economic variables. Descriptive and analytical statistics were obtained. It was found that, of the 3304 students, 5.9% (190) had a close relative who had died, 11.2% (363) a distant relative, and 19.8% (641) a friend. According to the multivariate analysis, those students who had a close family member who had died had greater depression (RPa: 1.48; CI 95%: 1.20–1.84; value p < 0.001) and stress (RPa: 1.41; CI 95%: 1.11–1.79; p value = 0.005), in addition, those who had a friend who died had higher levels of anxiety (RPa: 1.20; 95% CI: 1.06–1.36; p value =0.005); also, the respondents who suffered from COVID-19 had greater depression (RPa: 1.49; CI 95%: 1.05–2.11; value p = 0.024) and stress (RPa: 1.55; CI 95%: 1.05–2.28, p-value = 0.028). An association was found between suffering from depression, anxiety, or stress, and having suffered the death of a family member or close friend from COVID-19. This finding is an important one for places of education to consider, suggesting a need to generate psychological support programs for students who have lost a loved one during the pandemic, since this could have academic and social repercussions. An association was found between the three mental illnesses studied and the death of a family member or close friend from COVID-19.