Peter Breur said no to this question when it was asked in an exchange under another one of my questions "Why do physicists say...". His reason is this is not allowed by Einstein's E = mc^2. Who else agrees with him?
There is a reluctance to answer this question. The correct answer is YES. Special relativity is believed to not allow massive lightspeed particles because of an error made more than 100 years ago which has never been discovered and corrected until now. It has been hidden for more than 100 years because of the successful effort to cover it up because it violates Einstein's E = mc^2 and does not fit into his framework.Thus the present understanding that the Einstein relations apply to all particles became the accepted law of physics. This belief is wrong. Einstein's E = mc^2 applies only to atomic particles, that is to particles of type 2 that can't travel at the speed of light.
It is important to realize that there is nothing wrong with Special Relativity itself. There is only something wrong with its present understanding because that understanding contains a 100 year old error which I have identified and am attempting to alert everybody to, and remove. That is the main task of Photodynamics, see my paper Photodynamics in JPM 2018 Vol 9.
To all readers: Do you agree with me that special relativity, when corrected by the laws of Photodynamics, (which belong in SR) allows massive particles to travel at the speed of light?