Grading: (1) The chain is "hasan" on account of Muhammad bin Ishaq, because he explicitly stated that he heard it from al-'Abbas bin Ma'bad (al-'Abbas bin 'Abd Allah bin Ma'bad bin al-'Abbas).
Technical note / Defect ('illah): In this chain the words "from his father" are found only with the author (al-Hakim), whereas all the other students of Ibn Ishaq narrate "from some of his family, from Ibn 'Abbas".

Note / Clarification: Imam al-Bayhaqi (3/140) narrated it with al-Hakim's own chain, but there it is "from some of his family" instead of "from his father"; so it is surprising that al-Hakim specified it here by naming the father.

Technical note / Defect ('illah): Al-Mizzi said in "Tahdhib al-Kamal" (35/82-83) that the unnamed family member is either his father, or his brother Ibrahim, or 'Ikrimah the client of Ibn 'Abbas, and all of them are thiqah.
Important point: In light of a sahih narration in al-Bukhari's "al-Tarikh al-Awsat" (1/380), the more likely view is that the unnamed person here is "'Ikrimah, the client of Ibn 'Abbas". And Allah knows best.

Note / Clarification: This statement of Ibn 'Abbas about the martyrdom of Abu Hudhayfah at the Battle of al-Yamamah is narrated after the incident of the capture of al-'Abbas bin 'Abd al-Muttalib at the Battle of Badr, so it is fitting to reference it with the next narration.