  The Hammersley problem asks for the maximal number of points in a monotonous
path through a Poisson point process. It is exactly solvable and notoriously
known to belong to the KPZ universality class, with a cube-root scaling for the
fluctuations.Here we introduce and analyze a variant in which we impose a
Lipschitz condition on paths. Thanks to a coupling with the classical
Hammersley problem we observe that this variant is also exactly solvable. It
allows us to derive first and second orders asymptotics. It turns out that the
cube-root scaling only holds for certain choices of the Lipschitz constants.
