Most websites/references online seem to indicate that polar (as in polar bear, or polar opposite) is pronounced with an "er" phonogram ending (as in "her"). However a phonetic analysis of the word would seem to imply the word polar is made up of the phonographs "p" "o" "l" and "ar" (as in what a pirate says). Indeed from the Cambridge dictionary website the British pronunciation has the (pirate) "ar" phonogram enunciation. So which is correct? Is it a matter of preference?
If you're talking about this Cambridge Dictionary entry, it's pronounced with /ə/, like "her" (the transcription given is "UK /ˈpəʊ.lər/", and the audio is consistent with that transcription). So I'm puzzled by the statement that "from the Cambridge dictionary website the British pronunciation has the (pirate) "ar" phonogram enunciation."
There's nothing about the spelling of "polar" that makes the pronunciation with /ə/ unlikely. "Ar" commonly represents /ə(r)/ in unstressed syllables.
See the notes on the web page "Spelling the Vowels of English Received Pronunciation" (which seems to be based on date from one of the projects of Washington University in St. Louis’s Reading and Language Lab):
<a> is by far the most common spelling for /ə/ in the general case (saliva), but 8 or 9 other spellings are also quite common, including <o> (daffodil), <er> (clever), <e> (shellac), <or> (forbid), <u> (triumph), <ar> (cellar), <re> (fibre), <ur> (pursue).
You might be able to find some cases where people pronounce polar with /ɑː/, but neither the spelling nor common usage provide support for saying that it is "correct" to pronounce it with /ɑː/ instead of /ə/.