  We formulate the stringy nonrelativistic limits of the flat space and
AdS$_5\times$S$^5$ string as coset models, based on the string Bargmann and
extended string Newton-Hooke algebras respectively. Our construction mimics the
typical relativistic one, but differs in several interesting ways. Using our
coset formulation we give a Lax representation of the equations of motion of
both models.
