This study aimed at assessing the behaviour of Konik geldings and mares, kept in a stable and in a free-range system, during behavioural tests regarded as a determinant of the exploration urge. A total of 19 Konik horses kept in individual stables and in a free-range system were included in the study. The experiment was conducted in five phases separated by five-day breaks. A one-stage passive human test was performed during the first phase, a three-stage active human test—2nd phase, a three-stage unknown object test—3rd phase, a two-stage unknown surface test—4th phase, and a one-stage test of social isolation—5th phase. Ten attributes were analysed, including the horse sex and the keeping system. The results were also correlated with one another. Konik horses were found to show the urge to explore, although their behavioural responses are individual and stimulus dependent. In many cases, the horse sex and the keeping system influence the exploratory behaviour, although it is manifested by a greater intensity in geldings than in mares, and in free-range horses than in those kept in a stable. The study is regarded as preliminary due to the small number of horses in the study groups.