Casting instants between first strike/normal strike combat phases
Yes, that's a thing you can do.

It works out like this:


The first strike combat damage step begins:


All creatures with first strike or double strike assign combat damage. Your 1/1 deals its damage to the 4/4, which now has 1 damage marked on it.
All players get priority.
You cast a spell—Lightning Bolt—and deal 3 damage to the 4/4. It now has 4 damage marked on it and dies.
All players (including you) choose to do nothing else, passing priority to let the next step begin.

The regular combat damage step begins:


All creatures still present now deal damage, except your 1/1 because it already dealt its first strike damage. The 4/4 is no longer around, so it won't do anything.



Your 1/1 survives this combat.

Naturally we'd normally see this scenario if you can't deal a whole 4 damage with any one spell or creature, but you can combine these damage sources to get the job done. However, you're also definitely not limited to eliminating the creature via damage in the middle of combat: if you have a spell like Murder or Final Reward you can also remove the creature that way after first strike combat damage and have the same outcome. You may choose to do things that way if your 1/1 has Lifelink, to gain some life before eliminating the enemy creature.