I can’t find the option for it.
if you choose one of the “search for” items from left panel and you have dolphin set to remember your window state, then it will always come up that way when you open it.
But shall it remain visible whilst I browse? It appears to automatically close itself when it changes view.
you will need to use tabs to keep a search tab open if you want to see other views while search remains open.
As a workaround, you can keep it in a tab, or in a split view or a different dolphin window.