Aggressive geometry – Non-flat locus for smooth patterns

Let $ X $, $ Y $ to be connected smooth patterns of finite type on an algebraically closed characteristic field $ 0. Let $ f: X rightarrow Y $ to be a surjective non-birational morphism on the underlying topological spaces. Can the non-flat locus of $ f $ to be non-empty of codimension$ geq $ 2 in $ X $? For birational morphism, I think that ZMT plus a purity theorem show that the answer is "no".

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