Mbappe had the chance to level the scores at 1-1 in the 68th minute against Athletic, but the France international could not convert his spot kick, with the home side's goalkeeper Julen Agirrezabala making the save.
Jude Bellingham did level the scores in the 78th minute, but Gorka Guruzeta won it for the Basque outfit in the 80th minute, with Real Madrid losing for the fourth time in their last seven matches.
Mbappe also missed a penalty against Liverpool in the Champions League at the end of November, and there has been widespread criticism of his performances this season despite coming up with a respectable 10 goals in 20 appearances.
The 25-year-old took to social media after the clash with Athletic to apologise for his part in his side's latest defeat.
"Bad result. A big mistake in a match in which every detail counts. I take full responsibility for that. A difficult moment, but it's the best time to change the situation and show who I am," Mbappe posted on his Instagram story.
Mbappe admits he is currently in "a difficult moment"
Real Madrid allegedly ignored data surrounding Mbappe's physical condition before signing him on a free transfer over the summer, and he has struggled to make his mark as a centre-forward for Los Blancos this term.
Carlo Ancelotti's side remain second in the La Liga table despite their latest setback, four points behind leaders Barcelona with a game in hand, so they remain very much in the title picture at this stage of proceedings.
"I haven't spoken to him. I don't evaluate a player's performance based on a penalty, he's sad and disappointed but we have to keep going," Ancelotti told reporters when asked about Mbappe after the match.
"He's not at his best but we have to give him time to adapt. He can do better and he's working to do better."
Ancelotti opens up on "difficult game" against Athletic
Ancelotti branded Wednesday's clash with Athletic "a difficult game", but he has already switched his attention to Saturday's clash against Girona, which comes three days before a huge Champions League fixture away to Atalanta BC.
"It was a difficult game, it was close, very competitive and hard-fought. When we got the equaliser, we could have focused on controlling the game but fine details have cost us. We missed the penalty, but I can't be assessing the game on that basis because you might score, or not. Mbappe is upset, he's disappointed but we have to keep going," he said.
"It was an evenly matched game. Athletic are a very dangerous side at home. We competed well, fought hard and a draw would have been a fair result given how both teams performed. It was a solid defensive display from us in the first half, but we could have done a better job with the ball, we didn't create many chances.
"In the second half we looked a bit better and we had more opportunities. We're still in the mix, we could have got a more positive result tonight. We're going to keep working towards Saturday's game."
Real Madrid have won 10, drawn three and lost two of their 15 La Liga matches during the 2024-25 campaign, scoring 31 goals and conceding 13 in the process.
Written by
Matt Law