  A supersymmetric string model in the D=11 superspace maximally extended by
antisymmetric tensor bosonic coordinates, $\Sigma^{(528|32)}$, is proposed. It
possesses 30 $\kappa$-symmetries and 32 target space supersymmetries. The usual
preserved supersymmetry-$\kappa$-symmetry correspondence suggests that it
describes the excitations of a BPS state preserving all but two
supersymmetries. The model can also be formulated in any $\Sigma^{({n(n+1)\over
2}|n)}$ superspace, n=32 corresponding to D=11. It may also be treated as a
`higher--spin generalization' of the usual Green--Schwarz superstring. Although
the global symmetry of the model is a generalization of the super--Poincar\'e
group, ${\Sigma}^{({n(n+1)\over 2}|n)}\times\supset Sp(n)$, it may be
formulated in terms of constrained OSp(2n|1) orthosymplectic supertwistors. We
work out this supertwistor realization and its Hamiltonian dynamics.
  We also give the supersymmetric p-brane generalization of the model. In
particular, the $\Sigma^{(528|32)}$ supersymmetric membrane model describes
excitations of a 30/32 BPS state, as the $\Sigma^{(528|32)}$ supersymmetric
string does, while the supersymmetric 3-brane and 5-brane correspond,
respectively, to 28/32 and 24/32 BPS states.
