= Solution
No rectangle of positive width around $(0,0)$ has a uniform Lipschitz constant in $y$. For any nonzero $x$ in such a rectangle,
$$
\frac{|f(x,y)-f(x,0)|}{|y|}=|x|\,|y|^{-1/3}\longrightarrow\infty\qquad(y\to0).
$$
The vanishing factor $x$ at the single initial point does not control neighbouring $x$-values. The failure of the hypothesis is accompanied by actual nonuniqueness: both
$$
\boxed{y_1(x)=0,\qquad y_2(x)=\frac{x^6}{216}}
$$
satisfy the equation and have value zero at $x=0$. Direct differentiation gives $y_2'=x^5/36=x(y_2)^{2/3}$, including the zero point by continuity.
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