Brain-computer interfaces (BCIs) are tools for controlling computers and other devices without using muscular activity, employing user-controlled variations in signals recorded from the user's brain. One of the most efficient noninvasive BCIs is based on the P300 wave of the brain's response to stimuli and is therefore referred to as the P300 BCI. Many modifications of this BCI have been proposed to further improve the BCI's characteristics or to better adapt the BCI to various applications. However, in the original P300 BCI and in all of its modifications, the spatial positions of stimuli were fixed relative to each other, which can impose constraints on designing applications controlled by this BCI. We designed and tested a P300 BCI with stimuli presented on objects that were freely moving on a screen at a speed of 5.4°/s. Healthy participants practiced a game-like task with this BCI in either single-trial or triple-trial mode within four sessions. At each step, the participants were required to select one of nine moving objects. The mean online accuracy of BCI-based selection was 81% in the triple-trial mode and 65% in the single-trial mode. A relatively high P300 amplitude was observed in response to targets in most participants. Self-rated interest in the task was high and stable over the four sessions (the medians in the 1st/4th sessions were 79/84% and 76/71% in the groups practicing in the single-trial and triple-trial modes, respectively). We conclude that the movement of stimulus positions relative to each other may not prevent the efficient use of the P300 BCI by people controlling their gaze, e.g., in robotic devices and in video games.