Despite logic, the spelling of the noun for 'a native of the PHilippine [with one '-l-' and 3 '-p-'s] islands is Filipino, with only one '-p-'. You may properly, if a little pedantically, write the feminine adjective as filipina, as that is its form in Spanish. OED records occurrences of the adjective as spelled like the place-name. Inhabitants of the Phillipines call themselves Pinoy, their name in the most widely-spoken local language, Tagalog.
The adjective and noun for 'connected with, or native to, the islands' is taken from their name in Spanish, that is "(las Islas) Filipinas". The English version of the place-name the 'Philippine islands' is taken from the English spelling of the same name, which was originally Greek. 'Philip', or 'horse-lover' (phil[o]-hippo[s]) is spelled in Spanish according to the phonetic rules of that language as Felipe. Both the Spanish 'F-' and the English 'Ph-' are the standard representations, in their own languages, of the Greek letter φ, 'phi'.
- The first use of the name was by the Spanish explorer Ruy López de Villalobos during his expedition in 1542. He applied it to two of the largest islands to honour the then Prince of Asturias (~ heir apparent) Philip of Austria (1527-1598); the name was extended to the whole archipelago after he had ascended the throne as Philip II of Spain - or Felipe II in his own language (1556 - 1598). We can find no explanation supported by evidence for the divergence in spelling, in current practice, between the adjective in its Spanish form and the place-name in its English form. It may be that charts used by British sailors were printed in the English form, but no one wrote the adjective down till later. The earliest appearance of Filipino recorded in OED is 1898.