A non-relativistic proton linac based on high gradient backward travelling wave accelerating structures was designed using a novel dedicated 3D particle tracking code. Together with the specific RF design approach adopted, the choice of a 2.9985 GHz backward travelling wave (BTW) structure with 150° RF phase advance per cell was driven by the goal of reaching an accelerating gradient of 50 MV/m, which is more than twice that achieved so far. This choice dictated the need to develop a new code for tracking charged particles through travelling wave structures which were never used before in proton linacs. Nevertheless, the new code has the capability of tracking particles through any kind of accelerating structure, given its real and imaginary electromagnetic field map. This project opens a completely new field in the design of compact linacs for proton therapy, possibly leading to cost-effective and widespread single room facilities for cancer treatment