Examples of non-differentiable problems in economics
Here are four I could think of:


Leontief and Lexicographic functions, used in preferences or
production functions are non-differentiable.
Labor models often employ a discrete labor supply (work or don't
work, sometimes alongside the decision of how much or how hard to
work).
Housing models often employ a non-convex adjustment cost to ensure
that there is a discontinuity in the cost function for the choice of
housing.
Kinked objective functions, where at one or more points the
left-hand and right-hand limits  differ, are also common in
economics (e.g. optimal option exercise at maturity)