A book written in the 1960s, for which I'm counting books published in the 1960s, is another really broad category.
As always, I'll start with academic, or near-academic books
I'm always happy for an oportunity to recommend E.P. Thompson's Making of the English Working Class which was first published in 1963.
A book that continues to influence the conversation on fascism is Hannah Arendt's Eichmann in Jerusalem which began in the New Yorker magazine in 1963.
Some of Erving Goffman's most important works were published in the 1960s.
Jane Jacobs' Death and Life of the Great American Cities came out in 1961
And, I've only just recommended it in the last post, but Frantz Fanon's The Wretched of the Earth was first published in 1961.
The 60s was a time of landmark books that both influenced and were influenced by social movements.
Alex Haley's as-told-to Autobiography of Malcolm X was published in 1965.
James Baldwin's The Fire Next Time was first published in 1963.
Another landmark 1960s activist book is Rachel Carson's Silent Spring, first published in 1961.
Richard Farina's Been Down So Long It Looks Like Up To Me , published in 1966, is a 60s classic and one of my favorite novels from my teen years, though I haven't read it in quite a while.
This entry's official song is "Sympathy for the Devil," a 60s song inspired by Mikhail Bulgakov's The Master and Margarita which was actually written in Russian between 1928 and 1941, though the English translation could qualify as "written in the 60s."
Science fiction also went through a radical transformation in the 1960s. The landmark collection Dangerous Visions (edited by Harlan Ellison) came out in 1967. Several of Samuel R. Delany's more accessible books came out in the 1960s. Ursula LeGuin's The Left Hand of Darkness was first published in 1969.
There are many lists that you can peruse for important fiction and popular non-fiction of the 1960s. Lit Hub is always a great place to look. Here is their list of The 10 Books that Defined the 1960s which is followed by a lengthy list of additional books published in that decade.