Śliwa won the championship of Poland six times. In 1946, he won the first Polish Chess Championship after World War II in Sopot (5th POL-ch). In 1948, he took 3rd in Kraków (6th POL-ch; Kazimierz Makarczyk won). He won the Polish championship four consecutive times in 1951–1954. He won his last title at Wrocław 1960 (17th POL-ch).
In tournaments, Śliwa tied for 9-10th at Sopot 1951 (Ernő Gereben won). In 1952, he took 17th in Budapest (Paul Keres won). In 1954, he tied for 12-14th in Bucharest (Viktor Korchnoi won). His best achievement was 3rd, behind Luděk Pachman and László Szabó, at Mariánské Lázně (Marienbad) 1954 (zt). In 1955, he tied for 19th-21st in the Göteborg (interzonal), which David Bronstein won. In 1957, Śliwa tied for 2nd-4th with Oleg Neikirch and Alexander Matanović, behind Miroslav Filip in Sofia (zt). In 1959, he tied for 5-7th in Riga (Boris Spassky won). In 1962, he tied for 4-7th in Mariánské Lázně (Mark Taimanov won). In 1965, he tied for 12-14th in Belgrade (Milan Matulović won). In 1966, he took 9th in Polanica Zdrój (Vasily Smyslov won). In 1966, he tied for 9-10th in Tel Aviv (Svetozar Gligorić won).
One of Śliwa's most famous games is his win over Bronstein in the Immortal losing game.
- In 1952, at fourth board in the 10th Chess Olympiad in Helsinki (+7 –1 =4);
- In 1956, at first board in the 12th Chess Olympiad in Moscow (+6 –4 =6);
- In 1958, at first board in the 13th Chess Olympiad in Munich (+6 –6 =5);
- In 1960, at first board in the 14th Chess Olympiad in Leipzig (+5 –5 =6);
- In 1962, at first board in the 15th Chess Olympiad in Varna (+7 –4 =5);
- In 1964, at third board in the 16th Chess Olympiad in Tel Aviv (+6 –2 =6);
- In 1966, at fourth board in the 17th Chess Olympiad in Havana (+7 –3 =5).
He won the individual silver medal at Helsinki 1952.