Just from the length of the connector inside the function generator? I have never seen a problem from that, but then I haven't looked very carefully. 1 nanosecond is also pretty fast for most passive probes regardless of the signal source.
I didn't realise that there could be such a huge variation in rise time for function generators; not that it will matter for the applications I will use it for, but if I did want to create a waveform that had a rise time of less than 3.5ns so I can see this limit on my scope, what is the best way to do this?
I misread that. High speed CMOS logic gates will switch considerably faster than your function generator, but I think the 74AC series is still on the order of a few nanoseconds.
To go faster than 3.5 nanoseconds you have to move away from standard function generators. They simply aren't designed for that. There are some dedicated pulse generator instruments that can do it. Avalanche pulsers are one option, as are step recovery diodes. There are extremely fast logic families like ECL and newer standards like LVDS that can have really high toggle frequencies and correspondingly fast edges. ECL can have edge times <200 ps.
Needless to say, the faster you go, the harder everything gets. The source may be blazing fast, but the connection to your scope can be as big of a performance limit as the scope itself. At several GHz everything is a transmission line, and even a slight reflection can totally smear out your rise time.