This is a highly debatable topic.
There are many business owners who are highly successful, yet they have never had a business plan. Similarly, there are many businesses that have failed because they never had a plan.
There will inevitably be research for both arguments. However, my personal belief is that while there may be many successful businesses who did not have a business plan, I believe that the reality is that we are in fact referring to a WRITTEN business plan when this statement is made.
I think a written business plan may not be crucial, but I am sure that all successful businesses had a plan, albeit that it was perhaps merely a mental plan. Most successful owners I have met had a clear vision of where they were going. That to me is indicative of a plan.