The summer arrival from Liverpool has been ruled out for two months due to a thigh injury.
Taking to social media, Alexander-Arnold posted: "Absolutely devastated by this.
"The moment hurts but I'm going to work hard to come back stronger and better! See you in 2026 Madridistas."
The injury is a blow for Alexander-Arnold, who was just finding some form with Real. For the win at Athletic Bilbao, he created the first goal of Kylian Mbappé.
In the 55th minute he asked for a change as he suffered the setback.
What games Trent could miss:
Real Madrid – Celta de Vigo: December 7 (9:00 p.m.)
Real Madrid – Manchester City: December 10 (9:00 p.m.)
Alavés – Real Madrid: December 14 (9:00 p.m.)
Copa del Rey round of 32: December 16/17/18
Real Madrid – Seville: December 20 (9:00 p.m.)
Real Madrid – Betis: January 4 (4:15 p.m.)
Atlético de Madrid – Real Madrid (Super Cup): January 8
Possible final of the Spanish Super Cup: January 11
Copa del Rey round of 16: January 13/14/15
Real Madrid – Levante: January 17/18
Real Madrid – Monaco: January 20 (9:00 p.m.)
Villarreal – Real Madrid: January 24/25
Benfica – Real Madrid: January 28 (9:00 p.m.)
Real Madrid – Rayo Vallecano: January 31/February 1