Could the Missouri River be running while Lake Michigan was frozen several meters deep?

During an ice age, would it be possible for a great lake to be frozen while a relatively nearby river was still running? To help illustrate my point:

Would it be possible for Lake Michigan (or parts of it) to be frozen up to ten meters deep? All the while the Missouri River was still running around Kansas City? Or would all surrounding bodies of water necessarily be frozen solid?

Are such varying temperatures and conditions feasible in any way? What should be taken into account?

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Could the Missouri River be running while Lake Michigan was frozen several meters deep?

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