Did Gautama Buddha exist? Is there any historical third-party evidence that the Gautama Buddha actually existed?

Is there historical third-party evidence that Prince Siddhartha or the Gautama Buddha existed outside of the writings of Buddhism?
There are lot of archaeological evidence align with Buddha's life story. I think those can be considered as the most trusted third party evidence. 
Here I am listing some of the web search findings. 


  Archaeologists working in Nepal have uncovered evidence of a structure at the birthplace of the Buddha dating to the sixth century B.C. This is the first archaeological material linking the life of the Buddha — and thus the first flowering of Buddhism — to a specific century.


You could find the full story from here and here

Apart from that, king Ashoka built Pillars at Buddhist monasteries, many important sites from the life of the Buddha and places of pilgrimage. This was done around 200 -300 years after the Buddha parinirvana. During that time those important sites and monasteries should have left some sort of evidence to support Buddhas existence that convince King to build the pillers. So it also can be considered as good indication. 

Further:


  The records which we can find in the Buddhist countries where people received Buddhism a few hundred years after the Buddha's passing away such as Sri Lanka, Burma, China, Tibet, Nepal, Korea, Mongolia, Japan, Thailand, Vietnam, Cambodia and Laos show unbroken historical, cultural, religious, literary and traditional evidence that there was religious Teacher in India known as Gautama the Buddha. Ref: here