Radiated power by the antenna
As you've correctly surmised, the amount of power radiated from the antenna, physically, is 1 milliwatt. This makes sense since the antenna is lossless (100% efficiency), you've fed in 1 milliwatt, so by conservation of energy the antenna is radiating at 1 milliwatt.

What I think the question is means to ask for is the EIRP, or Effective Isotropic Radiated Power. The EIRP is a measure of much power a hypothetical isotropic (perfectly omnidirectional) antenna must radiate so that the power measured in the far field of the main lobe of the directional antenna is equal to the power of the isotropic antenna measured at the same distance. This is a useful figure for doing a link budget.

If the question is asking for EIRP, then 4 milliwatts is correct:

$$10log_{10}(1 mW) = 0 dBm$$
$$0 dBm + 6 dB = 6 dBm$$
$$10^\frac{6 dBm}{10} = 4 mW$$

If this is a homework assignment, you should write a note to your instructor. The question is ambiguous and should clearly state that it's looking for the EIRP.