Spain's largest cities — Madrid, Barcelona, Valencia, Seville and others — have access to round-the-clock veterinary emergency care, either through a dedicated 24-hour clinic or an emergency hospital that a regular vet will refer you to after hours. Madrid and Barcelona in particular have several centres offering genuine overnight and weekend emergency cover.
In smaller towns and rural areas — which cover a large share of Spain geographically, especially inland — round-the-clock coverage is less certain, and the local on-call arrangement described in the Emergency Vets article becomes the more relevant system. Ask locally as soon as you arrive, since availability can shift seasonally in popular holiday regions along the coast.
Given the cost of a genuine emergency, it's worth having a payment method ready that works smoothly for a larger, unplanned charge — some clinics expect payment at the time of treatment rather than billing afterward.