Probably not. If you had the T98 granny gear four speed, then yes. Otherwise no.
That said, there is a good chance that you can bolt a transmission from a 67 F100 into a 56 F100. My memory of the manual transmissions available for 67 1/2 tons were:
1 - 303 sychronized three speed with dual bolt pattern front flange
2 - T85 / R11 overdrive three speed with dual bolt pattern front flange
3 - the above mentioned T98 heavy duty four speed.
I think the inner bolt pattern on the dual pattern transmissions is the same as the bolt pattern on the original column shift three speed transmissions for 56. Drive shaft modifications would be required depending on what transmission you end up using.