Annoyingly technical answer: No, because it couldn’t have been. The “Pahlavi script” considerably postdates the Arabic script, in the sense that Pahlavi itself was not written down at all until well after the Arab conquest of Iran, probably in response to a newfound fear of the total loss of cultural and Zoroastrian religious material. In another sense, though, Pahlavi script is much older, since “Pahlavi script” is actually Aramaic script. Aramaic was a bureaucratic language of long standing, and you didn’t have to actually speak it to use it for your administration. The people who wrote down Pahlavi were trained in the use of bureaucratic Aramaic as the language of imperial administration under the Sassanids, and were so used to using it to write Aramaic that they actually spell words meant to be pronounced in Pahlavi with the Aramaic spelling of the equivalent word: if you’re reading a Pahlavi text and come across the word “MLK” (malka, king) you’re meant to pronounce it “shāh”. The Pahlavi script is just a very weird and late use of the Aramaic script.

So, now that I’ve picked that nit, is the Arabic script based on Aramaic, then? Yes, distantly, it is. Aramaic is the ancestor of all the abjad scripts of Africa and Asia, the scripts which primarily only include consonants and not vowels. Arabic doesn’t derive directly from Aramaic, but it does derive from Nabatean and related scripts which preceded it, and THOSE derive from Aramaic. The basic family tree of all these related alphabets is: Egyptian scripts > Phoenician abjad > Greek alphabet, Aramaic abjad, and others; Greek > Western scripts like Roman and Cyrillic; Aramaic > Eastern scripts like Hebrew, Arabic, Syriac and others. Wikipedia has a fun chart showing the correspondences of letters.

Thanks for this complex answer, especially the spelling adaptation was very interesting.