The use of drones in search and rescue (SAR) missions can be very cognitively demanding. Since high levels of cognitive workload can negatively affect human performance, there is a risk of compromising the mission and leading to failure with catastrophic outcomes. Therefore, cognitive workload monitoring is the key to prevent the rescuers from taking dangerous decisions. Due to the difficulties of gathering data during real SAR missions, we rely on virtual reality. In this work, we use a simulator to induce three levels of cognitive workload related to SAR missions with drones. To detect cognitive workload, we extract features from different physiological signals, such as electrocardiogram, respiration, skin temperature, and photoplethysmography. We propose a recursive feature elimination method that combines the use of both an eXtreme Gradient Boosting (XGBoost) algorithm and the SHapley Additive exPlanations (SHAP) score to select the more representative features. Moreover, we address both a binary and a three-class detection approaches. To this aim, we investigate the use of different machine-learning algorithms, such as XGBoost, random forest, decision tree, k-nearest neighbors, logistic regression, linear discriminant analysis, gaussian naïve bayes, and support vector machine. Our results show that on an unseen test set extracted from 24 volunteers, an XGBoost with 24 features for discrimination reaches an accuracy of 80.2% and 62.9% in order to detect two and three levels of cognitive workload, respectively. Finally, our results are open the doors to a fine grained cognitive workload detection in the field of SAR missions.