I harvested my first year’s growth of horseradish last fall, and let the smallest roots stay in the ground over the winter. This year I have a luxuriant growth of new small plants. They definitely spread very fast! I expect in 2-3 years they’ll be “invasive” hehe.
They are still in the same spot where they were last year, but since I did a crop rotation they are now in the “brassicas bed” instead of the “roots bed”.