Bhishma died on the bed of arrows after several days of completion of the MahAbhArata war.
Among Arjuna & Shikhandi, who can be considered the "slayer of Bhishma" or the reason of Bhishma getting killed ultimately?
When that lord of the world, the valiant Bhishma, having encountered Shikhandi, met with his death like a lion meeting with his at the hands of a jackal, what can it be but destiny? [Shalya Parva]
Then the slayer of Bhishma, the mighty Shikhandi, with all the Prabhadrakas, assailed the hero from every side with diverse kinds of weapons. [Sauptika Parva]
Moreover, it seems less likely that, Arjuna being devoted to Drona & Bhishma would have 'killed' them. This was evident during the discourse of Gita's 1st chapter. Since Bhishma didn't want to kill any of the PAndava, possibly Arjuna might have reciprocated the same.
As mentioned in this answer, Bhishma indeed would have informed about the way he can be killed to PAndava-s. However, ultimately Shikhandi would have pierced the majority of the arrows to Bhishma's body with some unspecified help from Arjuna as mentioned below (which interprets Arjuna as the slayer).
Viewing his relationship of grandsire unto them, the sons of Pandu, O sire, were spared in dreadful battle by Bhishma for ten successive days! Thyself also having laid aside thy weapons, the valiant Bhishma was slain in great battle by Phalguni with Shikhandi before him! [Karna Parva]