You can apply for a british passport at a british embassy or consulate.
Download and print the C1 form and C1 notes page, and fill out the C1 form. The C1 notes page is for your information, so keep that until your new passport has arrived. Those aged 15 or younger should use the C2 document set instead. The C1 and C2 document sets can be downloaded from the Foreign Office website,Take two new passport photos. These identical photos must be in color, no more than a month old, Draw up a cashier's check, certified check or money order in the amount necessary to pay your passport renewal fees, if you intend to pay by mail.
This should be made payable to the British Embassy Government Account, Gather additional documents. Submit the C1 or C2 application form, two passport photos,name-change relevant documents (if any) and the appropriate fee. You can do this either in person or by mail
CIVIL STATUS AND REGISTRATION OFFICE - Government office
Head of Civil Status & Registration: 200 51725
Senior Executive Officer: 200 51725
High Executive Officer: 200 51725/200 51726
Nationality Passport & Visa Section
Executive Officer: 200 51725 / 200 76945 / 200 51726
General Office: 200 51725 / 200 76945 / 200 51726
Fax: 200 42706
Email: passports.csro@gibraltar.gov.gi / visas.csro@gibraltar.gov.gi
Civil Status and Registration Office
Sir Joshua Hassan House
3 Secretarys Lane
Gibraltar
You can apply for a British passport if you have British nationality. But there are some circumstances where your application can be refused i.e;
Passports belong to the government and can be cancelled or withdrawn at any time.
Please attach documents that can be used by people. e.g. links
Please attach sample completed documents that would help other people.
It takes at least 6 weeks to get your first adult passport once your form has been received.
Videos explaining the procedure or to fill the applications. Attach videos using the following tag <&video type="website">video ID|width|height<&/video&> from external websites. Please remove the "&" inside the tags during implementation. Website = allocine, blip, dailymotion, facebook, gametrailers, googlevideo, html5, metacafe, myspace, revver, sevenload, viddler, vimeo, youku, youtube width = 560, height = 340, Video ID = Can be obtained from the URL of webpage where the video is displayed. e.g In the following url "http://www.youtube.com/watch?v=Y0US7oR_t3M" Video ID is "Y0US7oR_t3M".
Gibraltar is a British Overseas Territory. Gibraltar governs its own affairs, though some powers, such as defense and foreign relations, remain the responsibility of the UK Government. The Gibraltar passport is a British passport issued to Gibraltarians and other British residents in Gibraltar.
When you have completed the application form, section 5 should be completed and signed by a Member of the Gibraltar Parliament, Justice of the Peace, Minister of Religion, a professionally qualified person (for example, Doctor, Engineer, Lawyer, Teacher), Bank Officer, Established Civil Servant, Police Officer or a person of similar standing who has known you personally for at least two years, is a resident of Gibraltar and who is either a British Citizen, a British Overseas Territories citizen, British Overseas citizen, a British subject. A relative should not countersign.
http://www.ind.homeoffice.gov.uk/britishcitizenship/othernationality/britishoverseasterritories/
http://www.gibraltar.gov.gi
https://www.gibraltar.gov.gi/passports-and-visas
More information which might help people.