Karim Benzema and Cristiano Ronaldo both hit the post while Sergio Ramos was sent off as Real Madrid were held at Athletic Bilbao.
Zinedine Zidane’s champions needed a win to reduce the eight-point gap on leaders Barcelona, who drew 2-2 with Celta Vigo earlier in the day.
But Real were frustrated and efforts from Benzema and Ronaldo, with a clever volley, were as close as they came.
Real finished with 10 men after Ramos was dismissed for two bookings.
Ramos, dismissed for the 19th time in La Liga, had been booked for a foul on Raul Garcia when he caught Aritz Aduriz in the 86th minute and received his second yellow.
Real keeper Keylor Navas produced fine saves to keep out Inaki Williams and Aduriz.
This time last year Real were at the top of La Liga on 34 points from 14 games after Ramos’ last-minute equaliser at Barcelona in El Clasico.
The point in the Basque Country left them fourth in the table on 28 points from the same number of games.
Real’s next game is at home to Borussia Dortmund in the Champions League on Wednesday (19:45 GMT).