In term of SEO ranking, responsive website is not a mandatory but it's recommended. I've seen a lot of websites still rank well on Google without any responsive design on their websites.
On August 2015, Google’s John Mueller stated on his Google+ post regarding Google recommends responsive:
"A good way to make it [a site] work in both worlds [mobile & desktop] would be to have a site that uses responsive web-design techniques to adjust to the size of the user’s device/settings.
Still it leaves room for ambiguity. He only stated that it's recommended, but not clear sign (as usual) whether it's related to ranking factor or not.