Anatoly Bannik grew up in Kyiv, and was a childhood friend and chess rival of David Bronstein, later a world challenger.
Bannik made his high-level debut at age 18 in the 1940 Ukrainian Chess Championship in Kyiv, placing last in the field of 18 with 5.5/17.
At Vilnius 1949, he placed a fine fourth, with 11/17, behind joint winners Vladas Mikėnas, Semyon Furman, and Alexei Sokolsky.
He also won the Championship of the Spartak Club at Minsk 1962 with 12/17, ahead of Ratmir Kholmov, Alexander Zaitsev, and Alexei Suetin, who all scored 11/17.
Bannik moved to Germany later in his life, and was active in chess as late as in the year 2000, at age 79, in German team events.