Does a creature with double strike deal first strike damage and then regular damage in the same combat?

Double strike (CR 702.4) causes a creature to deal combat damage twice: once during the first strike combat damage step and once during the regular combat damage step. This means it gets the benefits of first strike (dealing damage before creatures without first or double strike) while also dealing its full power worth of damage again in the normal damage step.

CR 702.4b specifies that if at least one attacking or blocking creature has first strike or double strike, there is a first strike combat damage step (CR 702.4b, 510.1). Creatures with double strike then also participate in the regular combat damage step (step 510) along with creatures that have neither first nor double strike.

Importantly, if a creature without first or double strike is dealt lethal damage during the first strike damage step, it dies before it can deal damage in the regular step. A double striking creature can thus 'trade up' by killing a blocker early and surviving to deal its damage again.

Example: Your 2/2 creature with double strike attacks and is blocked by a 4/4. In the first strike damage step, your creature deals 2 damage to the 4/4. In the regular damage step, it deals 2 more damage, totaling 4 — enough to destroy the 4/4. Meanwhile, the 4/4 (lacking first or double strike) only deals its 4 damage once, during the regular step, killing your creature. Both creatures end up dying, but your creature dealt damage twice.

Note that if the defending creature is destroyed during the first strike step (e.g., it only had 2 toughness), your double striker would deal no further combat damage in the regular step since the blocker is gone — demonstrating the tactical power of double strike against smaller blockers.