Perhaps this is a stupid example: $n=2$, $m=1$, $Ax=x_1-x_2$ and $b=(3,1)$. For any $\lambda>0$ the minimizer $x^*$ is on the line segment between $b$ and $(2,2)$, hence all coordinates of $x^*$ are nonzero. So, any $\lambda>0$ is a "minimizer" no matter how large or small.
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