Background: The COVID-19 pandemic has contributed to the occurrence of work-related stress on nursing staff. Being resilience an essential element to countering adversity. The aim of the study was to assess burnout syndrome as well as resilience in hospital-care nurses during the first outbreak of the COVID-19 pandemic. Methods: A cross-sectional descriptive study of burnout syndrome and resilience of 101 nurses during the first COVID-19 outbreak. The Maslach Burnout Inventory and the Scale of Resilience of Connor-Davidson were used. Results: The burnout average score was 74.35 ± 12.78 points, and resilience was 27.94 ± 5.84. Temporary nurses reached a lower average score for the emotional fatigue dimension (23.80 ± 10.39 points) p < 0.05. The emotional fatigue dimension correlated adversely with the average score of resilience (r = −0.271; p < 0.001). Conclusions: The level of burnout in nurses was high, being higher on those who took care of COVID-19 patients. Resilient nurses were able to better cope with stressful situations.