This paper is devoted to the study of a semi-symmetric metric connection on an almost contact \(B\)-metric manifold. Recall that an almost contact manifold is a \((2n+1)\)-manifold admitting a vector-valued linear map \(\varphi\), a vector field \(\xi\) and a \(1\)-form \(\eta\) such that \[ \eta(\xi) = 1, \qquad \varphi^2 x = -x + \eta(x)\xi, \] for any vector field \(x\). If it additionally admits a pseudo-Riemannian metric \(g\) of signature \((n+1,n)\) satisfying \[ g(\varphi x, \varphi y) = -g(x,y) + \eta(x) \eta(y), \] then it is said to be an almost contact \(B\)-metric manifold. The authors deduce a relation between the Levi-Civita connection and the semi-symmetric metric connection on the considered manifold and determine the class of the torsion tensor corresponding to the semi-symmetric connection. Besides, they study Ricci-like solitons on almost contact \(B\)-metric manifolds with the considered semi-symmetric connection.