We present several results about a bilinear control problem governed by a semilinear elliptic equation, where the control acts as the coefficient in the reaction term. We first prove differentiability in \(L^2(\Omega)\), and weak-to-strong continuity, of the control-to-state mapping. This allows us to obtain existence of solution, as well as first order necessary optimality conditions and no-gap second order sufficient optimality conditions for the control problem. Next, we show how to state the problem using a semismooth equation and prove superlinear convergence of the semismooth Newton method in an infinite-dimensional framework. This is done under the assumptions of no-gap second order sufficient optimality conditions and strict non-complementarity, as is usual for this kind of algorithms in the finite-dimensional setting. Previous results in the literature assumed local convexity, which is a rather stringent assumption for bilinear control problems. An algorithm is provided. Numerical experiments that confirm all our findings will be presented.