  We develop a procedure for the complete computational enumeration of lattice
$3$-polytopes of width larger than one, up to any given number of lattice
points. We also implement an algorithm for doing this and enumerate those with
at most eleven lattice points (there are 216,453 of them).
  In order to achieve this we prove that if $P$ is a lattice 3-polytope of
width larger than one and with at least seven lattice points then it fits in
one of three categories that we call boxed, spiked and merged.
  Boxed polytopes have at most 11 lattice points; in particular they are
finitely many, and we enumerate them completely with computer help. Spiked
polytopes are infinitely many but admit a quite precise description (and
enumeration). Merged polytopes are computed as a union (merging) of two
polytopes of width larger than one and strictly smaller number of lattice
points.
