Last Nonzero Digits of a Factorial in Base

Mathematica, 57 48 bytes

Saved 9 bytes thanks to @2012rcampion .

IntegerString[#!/#2^#!~IntegerExponent~#2,##2]&

Python, 198 192 181 chars

def F(n,b,k):
 p=5820556928/8**b%8;z=0;e=f=x=1
 while n/p**e:z+=n/p**e;e+=1
 z/=1791568/4**b%4;B=b**(z+k)
 while x<=n:f=f*x%B;x+=1
 s='';f/=b**z
 while f:s=str(f%b)+s;f/=b
 return s

It's fast enough, ~23 seconds on the biggest example. And no factorial builtin (I'm looking at you, Mathematica!).

Pyth, 26 35 bytes

M?G%GHg/GHH.N>ju%g*GhHT^T+YslNN1T_Y

This is a function of 3 arguments, number, base, number of digits.

Demonstration.

The slowest test case, the final one, takes 15 seconds on my machine.