  A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the
unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends
to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called
$smooth$ if the unit ball has a unique supporting functional at each point of
the unit sphere. We prove that each non-smooth 2-dimensional Banach space has
the Mazur-Ulam property.
