Some of the Roman legions were extremely long lived (Ship of Theseus and all - the military tend to nurture their histories)[1]. The X Fretensis, for example existed for more then 400 years [2].
https://en.m.wikipedia.org/wiki/Honourable_Artillery_Company
Also, interesting or amusingly, the Coldstream Guards are the oldest regular army regiment, but originally it was raised by the roundheads in the civil war and fought the monarchy. Now it’s in the household division and used for royal protection.
In the order of precedence of the Foot Guards of the Household Division the Coldstream Guards are second after the Grenadier Guards. The latter's predecessor was founded in 1656 for the then still exiled Charles II.
The Coldstream Guards motto is Nulli Secundus - Second to None.