Born in Tel Aviv in 1924, Salzman began playing the piano as a young child and gave her first performance at age eight.
While on tour in Israel in 1932, Alfred Cortot heard her and invited her to study in Paris with him. Salzman also studied with Magda Tagliaferro at the Conservatoire national de Musique, where she won the Prix de Piano at age 14.
Violinist Bronislav Huberman, founder of the Israel Philharmonic Orchestra, attended one of her concerts in Paris and recommended that she be engaged as a soloist with the IPO. Her first concert in 1939 marked the beginning of long association with the orchestra: she frequently performed with them in Israel and also participated in two world tours, conducted by Carlo Maria Giulini and Josef Krips.
Salzman toured five continents several times, playing with major orchestras under conductors including Georg Solti, Antal Dorati, Malcolm Sargent, Igor Markevitch, Charles Munch, Zubin Mehta, Colin Davis, Walter Susskind, Andrew Davis and Neville Marriner. In 1963 she became the first Israeli musician invited to play in the USSR, and in 1994, the first Israeli pianist invited to play in China.
An enthusiastic chamber musician, Salzman was a member of the Israel Piano Quartet; she was also a regular guest with groups such as the Amadeus String Quartet.
She was head of the piano department at Tel Aviv University and served on the jury of many piano competitions including the Artur Rubinstein, Vladimir Horowitz, Iturbi/Valencia, Munich, Athens, Sydney and Seoul Competitions.
Pnina was the first recipient of the Israeli Prize for Music in 2006 and was active as a teacher and performer until last year.