  For any 1\leq p \leq \infty different from 2, we give examples of
non-commutative Lp spaces without the completely bounded approximation
property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3
these examples are the non-commutative Lp-spaces of the von Neumann algebra of
lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the
non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for
r large enough depending on p.
  We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have
the Approximation Property of Haagerup and Kraus. This provides examples of
exact C^*-algebras without the operator space approximation property.
