  Let $G$ be the symplectic group $Sp_4$ over a non Archimedean local field of
any characteristic. It is proved in this paper that for $p\in[1,4/3)\cup
(4,\infty]$ neither the group $G$ nor its lattices have the property of
approximation by Schur multipliers on Schatten $p$ class ($AP_{pcb}^{Schur}$)
of Lafforgue and de la Salle. As a consequence, for any lattice $\Gamma$ in
$G,$ the associated non-commutative $L^p$ space $L^p(L\Gamma)$ of its von
Neumann algebra $L(\Gamma)$ fails the operator space approximation property
(OAP) and completely bounded approximation property (CBAP) for $p\in[1,4/3)\cup
(4,\infty].$ Together with previous work [LdlS, HdL13a, HdL13b, dL], one can
conclude that lattices in a higher rank algebraic group over any local field do
not have the group approximation property (AP) of Haagerup and Kraus. It is
also shown that on some lattice $\Gamma$ in $Sp_4$ over some local field, the
constant function $1$ cannot be approximated by radial functions with bounded
(not necessarily completely bounded) Fourier multiplier norms on
$C^*_r(\Gamma)$, nor on $L^p(L\Gamma)$ for finite $p>4.$
