Yes. A pawn may checkmate a king in the right situation. Note, however, that technically speaking a king is never "killed" but is checkmated. The game is over before the pawn actually captures or "kills" the king.
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You can't "kill" the king with any piece. The king is the one piece that cannot be taken. The game is won when the king is in check and cannot move.
In that context, yes you can checkmate the enemy king with a pawn. How this is done depends on the position, but it's usually accomplished with the other surrounding pieces blocking the king's escape path.