  We construct a Lagrangian that describes the dynamics of a six-dimensional
free infinite (continuous) spin field in $6D$ Minkowski space. The Lagrangian
is formulated in the framework of the BRST approach to higher spin field theory
and is based on a system of constraints defining an irreducible representation
of the corresponding Poincar\'e group. The field realization of generators in
the $6D$ Poincar\'e algebra and the second-, fourth-, and sixth-order Casimir
operators are obtained in explicit form using additional spinor coordinates.
Specific aspects of such a realization in six dimensions are discussed. We
derive the conditions that determine the irreducible representation $6D$
infinite spin field and reformulate them as operators in the Fock space forming
a first-class algebra in terms of commutators. These operators are used to
construct the BRST charge and the corresponding Lagrangian. We prove that the
conditions of the irreducible representation are reproduced as the consequence
of the Lagrangian equations of motion, which finally provides the correctness
of the results obtained.
