  We classify lattice $3$-polytopes of width larger than one and with exactly
$6$ lattice points. We show that there are $74$ polytopes of width $2$, two
polytopes of width $3$, and none of larger width. We give explicit coordinates
for representatives of each class, together with other invariants such as their
oriented matroid (or order type) and volume vector. For example, according to
the number of interior points these $76$ polytopes divide into $23$ tetrahedra
with two interior points (clean tetrahedra), $49$ polytopes with one interior
point (the $49$ canonical three-polytopes with five boundary points previously
classified by Kasprzyk) and only $4$ hollow polytopes.
  We also give a complete classification of three-polytopes of width one with
$6$ lattice points. In terms of the oriented matroid of these six points, they
lie in eight infinite classes and twelve individual polytopes.
  Our motivation comes partly from the concept of distinct pair sum (or dps)
polytopes, which, in dimension $3$, can have at most $8$ lattice points. Among
the $74+2$ classes mentioned above, exactly $44 + 1$ are dps.
