The Dark Blues face an uphill task to avoid relegation after their latest Dens Park setback left them staring firmly at the prospect of returning to the First Division, but Smith was refusing to throw in the towel after seeing goals from Paul Lawson and Gary Glen kill their hopes of victory.
Smith's side now lie 14 points adrift of St Mirren having played a game more.
He said: "We'll never give up. Until it's mathematically impossible we will never say never and keep going for all the points we can get.
"It's becoming more difficult by the week but we just have to try and give ourselves any chance of staying up we can get.
"I don't know if their first goal was over the line or not after Rab Douglas got a hand to it. But the more annoying thing was the fact that their player was allowed to run so far without being challenged.
"We didn't play the way we know we can and certainly the last two games against Kilmarnock and Hibs."
Meanwhile, Ross County boss Derek Adams praised his on-form players as they moved into seventh in the table to almost erase any fears of relegation.
Adams said: "We're delighted with the end result. It's never easy coming to Dens Park and getting three points so all credit to the players.
"We're now only five points off second top and 18 points away from the bottom. It's a good night all round for the club. We've gone above Dundee United and Kilmarnock which is a massive boost."