Bottom-of-the-table Norwich gave their survival hopes a shot in the arm with last weekend's 2-0 win at home to Burnley but Smith's side were undone by a Cristiano Ronaldo hat-trick on Saturday in a 3-2 loss at Manchester United.
Smith was pleased with his side's performance but not the result and the Canaries stay seven points behind fourth-bottom Everton who have two games in hand.
Norwich are also 12 points adrift of fifth-bottom Leeds, both teams having just six games left.
Time is running out for the Carrow Road outfit who are no bigger than 1-100 to go down but as short as 1-500 with some firms but Smith is not throwing in the towel.
Asked if his side would just keep on pushing, Smith said at his pre-match press conference as quoted by Norfolk Live: "That’s what we will do.
"We can’t change today but we will hopefully carry on with the performance levels and the standards we’ve set today (against Man United) and last week against Burnley at home against Newcastle next week."