In a transistor, the relationship between the emitter current \( I_E \), the collector current \( I_C \), and the base current \( I_B \) is given by:
\[
I_E = I_C + I_B
\]
When the emitter current changes by \( \Delta I_E = 9.85 \, \text{mA} \) and the collector current changes by \( \Delta I_C = 9.5 \, \text{mA} \), we can find the change in the base current \( \Delta I_B \) using the following equation:
\[
\Delta I_B = \Delta I_E - \Delta I_C
\]
Substitute the given values:
\[
\Delta I_B = 9.85 \, \text{mA} - 9.5 \, \text{mA} = 0.35 \, \text{mA}
\]
Thus, the base current change is \( 0.35 \, \text{mA} \).
Therefore, the correct answer is option (4), \( 0.35 \, \text{mA} \).