One work around is to simply tap the end and use Loctite to lock in a socket head cap screw with a head less than 9 mm; 10-32 would be suitable.
If the shaft is not hardened you can use and Allen key as a broach to make a shallow hex.
You have to grind the end of a good quality Allen key perfectly flat to create a cutting edge.
Drill a hole about midway in diameter between the across flat dimension of the key and the across corner dimension in the end of the shaft about 1-1/2 Allen sizes deep, and drill a hole the same size as the across corner dimension about 1/2 the Allen key size deep.
Grip the shaft in the lathe chuck and the sharpened Allen key in a drill chuck in the tailstock.
Use the tailstock to push the key into the hole to broach the hex.
The larger hole acts as a guide hole making it easier to line things up.
If it is too difficult to push it in a do the broaching make the small hole larger; making this larger reduces the amount of hex created and reduces the amount of torque that can be applied to the shaft with an Allen in the hex; there is a compromise here between what it is possible to make and how well it works.
The Allen key of course will be tight in the hole it has broached so it is a good idea to use a pair of Imperial and Metric sizes that are close; use the large one for the broach and the other to turn the shaft. 7/32 is slightly larger than 5.5mm so that may be suitable for a 9mm shaft.