What it comes down to is that mass is not necessary for momentum. People are giving you equations (which are true, but don’t answer your question) or talking about relativistic mass (which is not something all physicists agree should be used.)
I find it helpful to remember that photons are literally waves of force. That force is in the electric and magnetic fields, but it is a moving packet of oomph that passes energy on to things it interacts with. That can create heat, yes, but remember that heat is just randomized kinetic energy.
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