AD8217 High Side Current Sensor Output at Low Currents
Did you use a Kelvin connection to connect the 5m\$ \Omega \$ to the AD8217?
If not, the measurement is not only the voltage drop across the resistor itself, but also across the contacts/traces as well.  



In the upper picture, the connection to the AD8217 has been made with the thin traces. However, the thick trace will have an impedance (resistance and inductance). Also the solder joints (or whatever connection you use) to the shunt resistor will have contact resistance.
In case of using 1\$ \Omega \$ or 0.3\$ \Omega \$ these contact resistance could probably be neglected or be incorrectly dismissed as being AD8217 offset voltage or bias current.
When using a 5m\$ \Omega \$ shunt resistor, you can certainly not neglect the above mentioned impedances.  

The impedances cause additional voltage drops (I represented them as \$V_{TC}\$), which is current dependent, which seems to match your findings. 

Using a Kelvin connection as shown in the lower picture will prevent measuring these voltage drops.