Lyrics is a tricky word. I've often seen the word lyric used in a singular form, particularly when referring to a particular line or a couplet of a song.
Around a century ago, in January 1917, Writer's Monthly chronicled the birth of a song called There's a Quaker Down in Quaker Town, using the singular form to refer to the title line:
Berg took the lyric to a number of other publishers, only to become a laughingstock. But again he went back to the Morris firm. This time he showed the lyric to Alfred Solomon, a composer, under contract to Morris. Solomon liked the lyric and wrote a melody for it. Then he offered the complete song to Morris.
In the modern era, Beach Boy Carl Wilson tosses around the words lyric and lyrics in an almost seemingly interchangeable way:
"On the first day of our collaboration, Brian played me the melody for a song he'd written. If I remember correctly, the original melody sounded exactly the way it does on the album, and someone had already written lyrics. Brian never played me the existing lyric -- he played the instrumental trace and said, 'I don't even want you to year the lyric that's been written." He ... then went to the piano and made a second tape, with him playing the melody and singing dummy lyrics. I took the cassettes from that first day home, and wrote the lyric to what became know as 'You Still Believe in Me'."
Source: "Wouldn't It Be Nice: Brian Wilson and the Making of the Beach Boys' Pet Sounds", Granata and Asher, 2016.
I would say that a song has lyrics, but we can refer to the lyric of a song, a term which often refers to a subset of the song's full lyrics.