Income tax

The basic rule is that you pay your tax in the country in which you perform the work which means in Spain. 

Furthermore to perform any work in Spain you must be a registered business of some sort or work for a Spanish arm of the company. 

That includes work performed wholly online, work is work.

But the employer is a company in UK and they do not have any branch in Spain.

I think you would benefit from professional advice.

But if you are Resident in Spain, you pay income tax in Spain (albeit there are a few exceptions, such as UK government pensions). By "Resident" I mean you have applied for and been granted Residency (you have the little green card) by meeting income, healthcare and home ownership/lease conditions.

If you don't have Residency, your options for staying here are constrained (and the constraints seem likely to tighten after Brexit, in whatever form that takes).

Many retired Residents here receive company pensions from firms without Spanish offices. They go through a process to switch their tax liability from Britain to Spain. I don't imagine that receiving a salary is much different.

The income tax take is generally higher in Spain than in the UK.

Than you will have to set yourself up as a business in Spain with your UK employer becoming a client whom you invoice.

If you become resident in Spain but carry on with your UK employment as if nothing had changed then as well as continuing to pay UK PAYE you will have to declare your gross salary in Spain which you will be taxed on IN IT'S ENTIRETY. 

In those circumstances the UK/Spain dual tax treaty will not apply because you will be flouting the rules. To not declare would be tax evasion which is severely dealt with.

Definitely seek professional advice but I think you'll find that what I've outlned is a reasonably fair and accurate synopsis.