How to reduce a MOSFET fall time with an IC driver
I'm using the IC \$\color{red}{\text{"ADP1071-2"}}\$ which provides isolation, and a lot of
  interesting features. I've been considering Vishay NMOSes for the
  switching transistor, and this being a flyback \$\color{red}{\text{(discontinuous)}}\$


The ADP1071-2 does not operate in \$\color{red}{\text{discontinuous}}\$ mode: -



Regarding the SiR692DP MOSFET, you might have problems reducing the turn-off or turn-on times: -



By the looks of your diagram you already have 30 ns fall time on the current and that looks about as good as you are going to get from this device. Having said that, you need to examine the data sheet and assure your self that the figures quoted are applicable to your specific design.

Looking at the data sheet and considering the MOSFET's output capacitance (\$C_{OSS}\$), I think at least 50% of the timing problem arises from this: -




When the device first turns off, \$V_{DS}\$ is around 0 volts and \$C_{OSS}\$ is over 3 nF
As \$V_{DS}\$ rises, \$C_{OSS}\$ does drop and probably accounts for 50% of the problem thereafter.