There aren't active "websites" dating back to 1986, if by "websites" we mean pages written in html and served up over http. But there are Internet domains that date back to the 1980s, specifically around 1985 or so (although you can argue that there are permanently assigned numerical IP addresses, as separate from human-readable domain names, that are older), and some of those domains now have websites associated with them.

To back up a bit: The Internet is not the same thing as the World Wide Web, although we often conflate the two of them.

The Internet is the worldwide network of interconnected computers that use the TCP/IP protocol to talk to one another, and which is the result of a merger of packet-switched networks that were created starting in the late 1960s, such as ARPANET, Cyclades, the Merit Net, and other networks.

The World Wide Web runs on the Internet, using HTML, CSS and Javascript to describe web pages, and using the hypertext transmission protocol (HTTP and its variants).

It's separate from other things that run on the internet, such as email, file transfer (FTP) and Usenet (NNTP).

The World Wide Web wasn't described until 1989 or public until 1990 -- an engineer at CERN, Tim Berners-Lee, wrote a proposal for information management and retrieval in 1989 and created HTML, the URI (later URL), and HTTP in 1990 (as well as the first web server and web browser).

Before Berners-Lee came up with the URI and later URL structure, there were already domain names on the Internet. Domain names grew out of computer addresses, which in the beginning were unique to each computer network. In the early days of networked computing, there were few enough network addresses that they could be kept manually -- Elizabeth Feinler and her team at Stanford kept a text file of host addresses for ARPANET many years, updating it as new hosts came online. (Users would literally call her up via telephone, she or her team would verify the information and add it to the hosts file.)

Over time, as ARPANET was split into military and civilian versions, and the number of hosts increased, the need for a system to keep domain names became more apparent, which led to the Domain Name System (DNS) that's in use today. DNS is a catchall term both for the database of names and the system used to describe them. Paul Mockapetris and John Postel, members of the Internet Engineering Task Force, promulgated DNS in 1985, which is when the earliest DNS addresses date to. This specification dates to 1987, but is still in use with some amendments today.

The DNS system describes domains at the top level -- things like .com, .mil., .org., .net -- as well as country codes, like .uk or .kz. Registration for domain names -- yoursite.com -- is handled by internet domain registrars, who sell those to the public. The oldest .com registration is to symbolics.com, registered in March 1985, although the oldest registration at all is the edu space registered to DARPA on January 1, 1985.