Rigid grippers used in existing aerial manipulators require precise positioning to achieve successful grasps and transmit large contact forces that may destabilize the drone. This limits the speed during grasping and prevents "dynamic grasping", where the drone attempts to grasp an object while moving. On the other hand, biological systems (e.g., birds) rely on compliant and soft parts to dampen contact forces and compensate for grasping inaccuracy, enabling impressive feats. This paper presents the first prototype of a soft drone -- a quadrotor where traditional (i.e., rigid) landing gears are replaced with a soft tendon-actuated gripper to enable aggressive grasping. We provide three key contributions. First, we describe our soft drone prototype, including electro-mechanical design, software infrastructure, and fabrication. Second, we review the set of algorithms we use for trajectory optimization and control of the drone and the soft gripper; the algorithms combine state-of-the-art techniques for quadrotor control (i.e., an adaptive geometric controller) with advanced soft robotics models (i.e., a quasi-static finite element model). Finally, we evaluate our soft drone in physics simulations (using SOFA and Unity) and in real tests in a motion-capture room. Our drone is able to dynamically grasp objects of unknown shape where baseline approaches fail. Our physical prototype ensures consistent performance, achieving 91.7% successful grasps across 23 trials. We showcase dynamic grasping results in the video attachment. Video Attachment: https://youtu.be/mqbj8mEyCdk