  In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for
$SL_3$ over a non archimedean local field $F.$ In this paper, we extend his
results to $Sp_4$ and therefore to any connected almost $F$-simple algebraic
group with $F$-split rank $\geq 2.$ As applications, the family of expanders
constructed from any lattice of such a group do not admit a uniform embedding
into any Banach space of type $>1,$ and any isometric affine action of such a
group, or its cocompact lattice, on a Banach space of type $>1$ has a fixed
point.
