This excerpt is taken from Dr. Yusuf al-Qaradawi's book Halal and Haram in Islam, translated into Urdu by Muhammad Tahir Naqqash.
Types of Permissible Games
There are many types of entertainment and numerous small forms of games that the Prophet Muhammad (peace be upon him) has declared permissible for Muslims so that they may lighten their burdens and have means of amusement. These games encourage a person to perform acts of worship and obligations, and to actively participate in them. Moreover, through these physical games, one gains training that increases strength and prepares a person for the battlefield of jihad. The following games belong to this category:
Competition in Running
The Companions, may Allah be pleased with them, used to compete in running, and the Prophet, peace be upon him, allowed them to do so. It is narrated that Sayyidina Ali, may Allah be pleased with him, was very fast in running.
Reference: Sunan Dar Qutni: 3054, Bayhaqi: 22/10 with a weak chain of narration
The Prophet, peace be upon him, himself competed in running with his pure wife Sayyida Aisha, may Allah be pleased with her, to please her and to teach the Companions. Sayyida Aisha, may Allah be pleased with her, says: The Messenger of Allah, peace be upon him, competed with me in running, and I got ahead. Then when my body grew, the Prophet, peace be upon him, defeated me in the race and said: This is the repayment for that time.
Reference: Musnad Ahmad: 2/26439, Abu Dawood, Book of Jihad, Chapter on racing against a man, Hadith: 2578. Also narrated by Ibn Majah in the Book of Marriage: 1979, very briefly
Wrestling
The Prophet ﷺ wrestled with the famous wrestler Rukana رضي الله عنه and threw him down several times.
Reference: Abu Dawood, Book of Clothing, Chapter on Turbans, Hadith: 4078; Tirmidhi, Book of Clothing, Chapter on Turbans on Helmets, Hadith: 1784
The jurists have derived from these ahadith the ruling that competing in running is permissible, whether men run against each other or a mahram runs with women or with their wives. This competition in running and wrestling does not contradict dignity, honor, knowledge, virtue, or old age in any way, because when the Prophet ﷺ ran with Lady Aisha رضي الله عنها, his age was more than fifty.