If a unit is on a battlefield, can you equip an equipment to it?

Yes, you can equip an equipment to a unit that is on a battlefield.

According to Rule 744.1.b, the Equip ability allows you to attach the equipment to a chosen unit you control. There is no restriction in the rules stating that the unit must be at your base; as long as the unit is under your control, it is a valid target for the Equip ability.

Important Timing Restrictions: While you can target a unit on a battlefield, remember that the Equip ability itself has specific timing restrictions:

• You must be in your turn.
• The Chain must be empty.
• There must be no Showdown occurring.

As long as you meet these timing requirements, you can equip a unit regardless of whether it is at your base or on a battlefield.