Keeper-batsman Matthew Wade takes over as skipper, with uncapped Queensland batsman Sam Heazlett joining the squad.
Smith, 27, was injured while fielding in Australia’s ODI win over Pakistan in the last match of their 4-1 series win.
Cricket Australia says Smith will have “scans to confirm the diagnosis” and is likely to “take 7-10 days to recover”.
Australia play the Black Caps on 30 January as well as on 3 and 5 February.
“It’s a bit of a shock,” said Smith. “I just landed really awkwardly on my foot and I felt something straight away.
“I was a little bit concerned, I’ve never really injured my ankle like that.”