McDonald’s Corp. said it plans to appeal after losing some European Union trademark rights to the Big Mac name following a challenge by smaller Irish rival Supermac’s.

The EU’s Intellectual Property Office in a Jan. 11 decision said McDonald’s failed to prove “genuine use” of the trademark across the EU over a continuous five-year period. The Irish fast-food chain filed its challenge in early 2017, the same year that the Big Mac celebrated its 50th anniversary.

The EUIPO, based in Alicante, Spain, agreed with Supermac’s that the evidence given by McDonald’s was “insufficient to prove that the EUTM was put...