Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, and Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field Fp. First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field F2and determine their weight distributions. Next, for any odd primep, we construct two classes of self-orthogonal linear codes fromp-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.