Single semiflexible polymer chains confined in a planar slit geometry between parallel nonadsorbing repulsive walls a distance D apart are studied by Monte Carlo simulations of a lattice model, for the case of good solvent conditions. The polymers are modeled as self-avoiding walks on the simple cubic lattice, where every 90 degrees kink requires a bending energy epsilon(b). For small q(b) = exp(-epsilon(b)/k(B)T) the model has a large persistence length l(p) (given by l(p) approximate to 1/(4q(b)) in the bulk three-dimensional dilute solution, in units of the lattice spacing). Unlike the popular Kratky-Porod model of worm-like chains, this model takes both excluded volume into account and approximates the fact that bond angles between subsequent carbon-carbon bonds of real chains are (almost) restricted to large nonzero values, and the persistence length is controlled by torsional potentials. So the typical local conformation in the model is a straight sequence of (on average) l(p) bonds (roughly corresponding e.g. to an all-trans sequence of an alkane chain) followed by a 90 degrees kink. While under weak confinement (D >> l(p)) the model (for very long chains) still is compatible with the Daoud-de Gennes scaling theory, for strong confinement. (D <= l(p)) strong deviations from the predictions based on the Kratky-Porod model are found.