According to this Barclays Bank Benidorm, banks in Alicante Spain. Services and location
there's a Barclays bank in Benidorm, but i don't know if it's up to date.
The easiest thing to do is Google name of bank Benidorm Spain That's how I found Barclays info. Apparently there are a lot of Lloyds banks dotted around Spain, but I don't think there's one in Beni.
It depends what you understand under the term english bank. If you want an established english bank, that has a counterpart here in spain, you can choose Halifax or Barclays. I would not recommend Santander, they might own Abbey but those banks are fully independent and not connected they don't offer special services like free transfer from the UK to Spain like Halifax does.
If you just want english speaking service you will find that most banks in expat heavy areas like Benirdorm, Calpe etc. actually have english and german speaking staff most of the time. I am with La Caixa, who even print your contract in german or english - even that they are a traditional spanish bank in our region.
As a last option there is always Solbank, which kinda "specialise" on english speakers and offer a full service which can include post boxes and translation services.
I would point out that if you ask Barclays in Spain or the UK about their counterparts they will tell you that they are completely separate entities. So whether you can call them a UK bank, I'm not sure.
Why do you say "British Bank" .... is it because you want English speakers?