Ligh does exert pressure. The fact that electroamgnetic waves (light being a form of electromagentic waves) exerts a pressure upon any surface exposed to it was deduced theoretically by James Clerk Maxell in 1871, and proven experimentally by Lebedev in 1900.
"Radiation pressure" is the pressure exerted upon any surface exposed to electromagnetic radiation. If absorbed, the pressure is the power flux density divided by the speed of light. If the radiation is totally reflected, the radiation pressure is doubled. For example, the radiation of the Sun at the Earth has a power flux density of 1,370 W/m2, so the radiation pressure is 4.6 uPa (absorbed), i.e. 4.6 x 10^-6 N/m2
Take another example, a 100 watt incandescent light bulb. Assuming the efficiency of of the bulb is 3%, i.e. only 3% of the power consumed by the bulb turn into visible light (the rest appears as heat). The power density at a distance of 1 metre from the bulb = 3/(4 x pi) W/m2 (where pi = 3.14159 ..... ), which is equal to 0.24 W/m2
The pressure exerted on a surface by the incident light from a 100-watt light bulb = 0.24/3x10^8 Pa = 8 x 10^-10 Pa = 0.8 nPa
Surely, theis is a very very small pressure compared with that from the atmosphere (100 000 Pa)