Towards p-adic string in constant B-field

Spacetime properties of the tachyon of the p-adic string theory can be derived from a (non-local) action on the p-adic line Qp, thought of as the boundary of the 'worldsheet'. We show that a term corresponding to the background of the antisymmetric second rank tensor field B can be added to this action. We examine the consequences of this term, in particular, its relation to a non-commutative deformation of the effective theory of the p-adic tachyon in spacetime