Introduction: Fear of infection with SARS-CoV-2 has become widespread. All over the world, since the very beginning of the pandemic, older adults have been considered one of the groups at highest risk of SARS-CoV-2 infection and death due to COVID-19. The aim of the study was to evaluate the severity of anxiety symptoms related to COVID-19 in the older adults who are participants of the Universities of the Third Age in Poland. Material and methods: The study included participants of the University of the Third Age in Poland. A total of 296 persons were enrolled, including 258 women and 38 men. The study was a diagnostic survey, conducted with the use of the following validated psychometric scales: General Anxiety Disorder-7 (GAD-7), Short Health Anxiety Inventory (SHAI), and State-Trait Anxiety Inventory (STAI). Results: In two scales (STAI and SHAI), the mean scores demonstrated mild symptoms indicative of anxiety disorders in the older respondents. Women and men did differ significantly in terms of the scores obtained in STAI X-1 and STAI X-2. Single respondents differed significantly from divorced ones in terms of STAI X-1 scores. Moreover, widows/widowers differed significantly from divorced ones in terms of STAI X-2, and GAD-7 scores. Respondents declaring their financial status as average differed significantly from those declaring their financial status as good in terms of: STAI X-1, STAI X-2, SHAI, and GAD-7 scores. Conclusions: The subjective experience of anxiety symptoms associated with fear of contracting COVID-19 was increased due to the ongoing pandemic, but was not significantly high in the analysed population of older people. COVID-19-related anxiety was significantly more common in lonely individuals and in those of worse financial status. Women and men differed significantly in terms of perceived state anxiety and trait anxiety measured by STAI. More studies addressing COVID-19-related anxiety in older people participating in the Polish Universities of the Third Age are needed to determine a more accurate distribution of this phenomenon in Poland.