The Gentlemen Pensioners were the forerunner to the modern Her Majesty's Body Guards. Henry VIII instated it in 1509, disbanded it in 1515/16, and re-created the unit in 1539.
They are described as the King's elite bodyguard (and here), but there is hardly any mention of them in public sources (most mentions seemed to refer to 18th/19th century fiction characters with a (a la 'The Gentleman Pensioner: A Romance of the Year 1569'). Their WP entry is nigh on useless for detail though it does mention two battles, one of which falls into the time when the aforementioned article says they were disbanded. WP does say this:
As the "Nearest Guard" to the Monarch, the unit attracted an aristocratic and aspiring membership, which could be utilised as a cadre of young officers when levies were raised for overseas service.
It also describes how the Gentlemen Pensioners were armed with battleaxes. Alison Weir describes their armament as poleaxes (talking about Mary Boleyn's husband William Stafford whose WP entry makes no mention of him belonging to the Gentlemen Pensioners). Weir's book also described how Stafford was meant to sleep (when on duty) in front of the King's bed/bedchamber, and to protect him from assassins.
WP also mentioned they partook of the Battle of the Spurs, but no specific mention is given.
The article above focussed on the Attempted Coup of July 1553 where many Gentlemen Pensioners sided with Northumberland (because they owed their positions is a simplified description) but there's no specific mention of combat, or of Mary being saved by the Pensioners who sided with her.
I should note that especially with Walsingham's prominence in the later Elizabethan court, a need for such a bodyguard (who sleeps on one's doorstep) seems to lessen slightly -- but I think this could be based on false assumptions (that Walsingham's spy ring would cut off all would-be assassins). In any case, Elizabeth kept the guard around so she must have seen value -- while Henry VIII clearly didn't think they were good enough value in 1515/16 if he actually disbanded them then.
Therefore, are there any specific recorded incidents where the Gentlemen Pensioners managed to fulfill their role as a bodyguard in the 16th century?