  We investigate Baire classes of strongly affine mappings with values in
Fr\'echet spaces. We show, in particular, that the validity of the
vector-valued Mokobodzki's result on affine functions of the first Baire class
is related to the approximation property of the range space. We further extend
several results known for scalar functions on Choquet simplices or on dual
balls of $L_1$-preduals to the vector-valued case. This concerns, in
particular, affine classes of strongly affine Baire mappings, the abstract
Dirichlet problem and the weak Dirichlet problem for Baire mappings. Some of
these results have weaker conclusions than their scalar versions. We also
establish an affine version of the Jayne-Rogers selection theorem.
