No. Modern-day mammography only involves a tiny amount of radiation — even less than that used in a standard chest X-ray.
Radiation is measured in rads. To give you an idea of just how low radiation exposure is during a mammogram, consider the following comparison. The views (individual x-ray shots) taken during a mammogram (usually 2 are taken) are around 0.1 - 0.2 rads. By comparison, cancer patients undergoing radiation treatment often receive anywhere from 1000 to 7000 or more rads in a normal course of therapy.