Can Spanish police operate in Gibraltar?

Short answer

Only within limits set by the Treaty. Spanish border officers carry out Schengen checks at Gibraltar airport, and any joint operation on Gibraltar soil must include Gibraltar officials, while policing in Gibraltar remains the responsibility of the Royal Gibraltar Police.

Detailed answer

The Treaty allows cross-border surveillance with prior notice and hot pursuit on the Schengen model. Pursuing officers must inform the local authorities, may carry service weapons for self-defence only, and may detain a person until local officers arrive without making an arrest.

All of these arrangements are reciprocal, so Gibraltar officers have the same powers in Spain, and anyone pursued into Gibraltar falls under the Gibraltar justice system.