Oblique droplet impacts onto a smooth surface at various inclination angles and at different ambient gas pressures were investigated using high-speed photography. It was found that the droplet splash can be entirely suppressed either by increasing the inclination angle or by reducing the ambient pressure. Variations of the threshold angle required for the splash suppression as a function of the impact velocity were determined, as well as the threshold pressure as a function of the inclination angle and the impact velocity. The threshold pressure increases monotonically as the inclination angle increases for small enough impact velocities but varies in a nonmonotonic manner for high enough impact velocities. Modifications of the existing splash model permit the theoretical determination of the splash threshold conditions that agree well with the experimental observations. It is shown that it is the velocity of the lamella tip that determines the splash onset.