The 6spd mechs will do 9 quite happily but as above not in that frame if you start stretching stays too far.
It is feasible but with work.
if the existing hub is Uniglide, the freehub body can be replaced with an 8spd but you'd have to some surgery to make room or as mentioned, the chain will rub on the stays or the final sprocket will foul the frame.
The back end may need widening a few mm to 130 spacing at minimum, the hub will need redishing to move the rim over and centralise it. but will make for a weaker wheel too.
Lots of tea, biscuits, 3in1 and time should see something emerge.