Why does a photon have linear momentum, although it doesn't possess mass?
The formula for the momentum, $p = mv$ is only valid in the context of classical mechanics. For photons, energy and momentum are related by $E = pc$, which, because $E = hv = \frac{hc}{\lambda}$ leads to the following formula: $p = \frac{h}{\lambda}$. So the momentum is always non-zero and depends only on the wavelength.

See this Wikipedia page for more details.