Application of the $p$-adic analysis and Green function in relativistic quantum theory is considered. Integrability of the free relativistic particle model is discussed. A convenient gauge is fixed. The Green function, as a corresponding propagator, are calculated in 4-dimensional $p$-adic Minkowski space. For this model, detailed calculation is performed for all three different cases: $p\equiv1(\operatorname{mod})4$, $p\equiv 3(\operatorname{mod})4$ and $p=2$.