Answer: The white or common mulberry we typically see is Morus alba and it is originally from China; the fruit however on mulberries of all kinds is a drupe or what looks like a berry, so there would not be any pods such as you are describing.
There is also the small tree or shrub called a Chinese Mulberry (Cudrania tricuspidata also called Cudrania triloba and Maclura tricuspidata) but it too produces a fruit rather than a long seed pod such as you described.
There is a Chinese Catalpa with long pods such as you described, Catalpa ovata, but it would be rather a rare tree in this country.
So, in answer to your question, maybe yes ... and no.
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