Can you give me a concrete example for a quadratic form $$ f(x,y)=ax^2+bxy+cy^2 \in \mathbb Z_2[x,y] $$ which has a primitive solution $(x^*,y^*) \in \mathbb Z_2 \times \mathbb Z_2$ (mod 4) with the property $$\frac{\partial f}{\partial x}(x^*,y^*) \not\equiv 0 \text{ (mod 4)},$$ but $f$ has no solution in $\mathbb Z_2 \ $? So, Hensel's lemma fails in this case.
Note: Hensel's lemma requires a solution (mod 8) for lifting.