Malinga, who missed practice Tuesday, trained at the Grenada National Stadium here Wednesday and bowled just three deliveries at reduced pace. He was then seen sitting in the stand with his left ankle heavily strapped.
Afterwards Sri Lanka team manager Michael Tissera told AFP: "He won't play tomorrow (Thursday) He's very keen to play but we've told him 'no'. He's torn a ligament.
"He suffered the injury the day before yesterday (Monday) and we sent him to Jamiaca for a scan because we couldn't do it here.
"Hopefully, he will be fit for the Australia or Ireland games (on April 16 and 18 in Grenada respectively)."
Malinga, whose distinctive slingy action makes him difficult to face, tops the tournament bowling averages with 15 wickets at an average of under 15 apiece.
Against South Africa, in a Super Eights match in Guyana last month, he became the first bowler to take four wickets in four balls in an international fixture.
Farveez Maharoof or Nuwan Kulasekara, both right-arm ace bowlers, are the leading contenders to take the 23-year-old Malinga's place in the side.
New Zealand, yet to win the World Cup, top the Super Eights table on run-rate ahead of champions Australia - the only other unbeaten side in the tournament.
Sri Lanka, the 1996 World Cup winners, are third in a table where the top four will qualify for the semi-finals.
The Super Eights conclude on April 21 with the World Cup final in Barbados on April 28.