Light is composed of photons so we could ask if the photon has mass. Even before it was known that light is composed of photons it was known that light carries momentum and will exert a pressure on a surface. This is not evidence that it has mass since momentum can exist without mass.
Relativistic mass is a measure of the energy E of a particle which changes with velocity.
By convention relativistic mass is not usually called the mass of a particle in contemporary physics so it is wrong to say the photon has mass in this way. In modern terminology the mass of an object is its invariant mass which is zero for a photon. By this definition a beam of light, is massless like the photons it is composed of. However, if light is trapped in a box with perfect mirrors so the photons are continually reflected back and forth in the box, then the total momentum is zero in the boxes frame of reference but the energy is not. Therefore the light adds a small contribution to the mass of the box.