Kinetic-theory analytical results concerning the thermophoretic velocity of a spherical nonevaporating or evaporating particle suspended in a high-temperature diatomic gas with appreciable dissociation degree (e.g., for oxygen with temperatures greater than 3000 K or for nitrogen with temperatures greater than 5500 K) are presented for the free-molecule regime. Molecular dissociation in the bulk gas and atomic recombination at the surface of the cold particle are included in the analysis. It is shown that the thermophoretic velocity of the suspended particle is directly proportional to the temperature gradient and approximately inversely proportional to the gas pressure. The thermophoretic velocities of both nonevaporating and evaporating particles are independent of the particle radius and increase slightly with increase in the specular-reflection fraction. For a nonevaporating particle, the thermophoretic velocity almost does not depend on the recombination fraction of atoms at the particle surface. For an intensely evaporating particle, the thermophoretic velocity (U-TV) increases with increasing thermal accommodation factor (a) and decreases with increasing atomic recombination fraction (alpha) at high gas temperatures with appreciable molecular dissociation, while U-TV almost does not depend on a and alpha at low gas temperatures.