Well I couldn't find any authentic hadith about the matter, even if among Muslim people the saying that Arabic would be the language of people in Jannah is well known and a kind of consensus. But there's no sahih hadith or Verse in the Qur'an which clearly supports this opinion.
On the other Hand there are ahadith most of them quoted by Imam at-Tabarni and qualified as weak or fabricated which support this opinion:
It was narrated by al-Tabaraani in al-Awsat, al-Haakim, al-Bayhaqi in Shu’ab al-Eemaan and others that Ibn ‘Abbaas (may Allaah be pleased with him) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Love the Arabs for three reasons, because I am an Arab, the Qur’aan is Arabic and the speech of the people of Paradise is Arabic.”
(Qualified as fabricated by ibn al-Jawzi and Imam ad-Dahabi)
Al-Tabaraani narrated in al-Awsat that Abu Hurayrah (may Allaah be pleased with him) said: the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “I am an Arab, and the Qur’aan is Arabic, and the language of the people of Paradise is Arabic.”
(Qualified as fabricated by al-Albani)
There are similar versions of this last Hadith but as ahadith marfu' (a saying of sahabi or tabi'a) narrated by ibn 'Abbas and Imam az-Zuhari which was quoted in Hadi al arwah of ibn al-Qayyim!
Imam Ibn Kathir pretended in his tafisri of the Verse in surat ash-Shu'ara' (26:195)
in plain Arabic language بلسان عربي مبين
that the language of the day of Qiyamah would be Syrianic and those who enter Jannah will speak Arabic (Fatwa in Arabic)
As there doesn't seem to be any clear evidence or reference telling us in which language people will speak neither in Paradise nor Hell nor in the Qiyamah we should accept the conclusion of Imam ibn Taymiyah in his Majmo' al-Fatawa (4/299): There is no basis for any of these ideas, whether on the grounds of common sense or in any report or text, rather they are mere claims that are devoid of any evidence. And Allaah knows best and is most Wise. (See this Fatwa for more details)