Previous studies mainly focused on the $p$-adic Potts model with countable spin values, demonstrating that this model has only one $p$-adic Gibbs measure. Furthermore, it was shown that the model exhibits a phase transition in the set of generalized Gibbs measures. A challenge remained to find a countable spin $p$-adic model where the set of all $p$-adic Gibbs measures would include at least two elements. In this talk we discuss the one-dimensional $p$-adic SOS model and demonstrated that the set of all $p$-adic Gibbs measures has continuum cardinality.