The description of Shielded Arm reads (emphasis mine):
Reinforcing veins of ore run through the target's arm, letting it ward off blows with its bare skin. It can use the Raise a Shield action to instead raise its arm, gaining a +2 circumstance bonus to AC. It can Shield Block with its Raised arm as well; when it does, the target reduces the damage as if it had a shield with Hardness 4 and 15 Hit Points. This shield doesn't have a Broken Threshold, and the spell ends if the shield's Hit Points are expended.
This spell doesn't modify the target's unarmed attacks and can't be used to make a shield bash Strike. Casting or coming under the effects of this spell also counts as using a metallic item with regards to anathema.
Unlike the similar Shield spell, the description doesn't specify whether or not a free hand is required.
The Raise a Shield action does also not specify the need to have a free hand.
You position your shield to protect yourself. When you have Raised a Shield, you gain its listed circumstance bonus to AC. Your shield remains raised until the start of your next turn.
Neither does the Shield Block feat:
You snap your shield in place to ward off a blow. Your shield prevents you from taking an amount of damage up to the shield’s Hardness. You and the shield each take any remaining damage, possibly breaking or destroying the shield.
Wielding a shield does require a free hand of course, but the point of this spell is that you don't wield a shield.
Does Shielded Arm require a free hand to raise and/or block?
Best Answer
No Free Hand Required
As you've noted nothing in the spell's description or the Raise a Shield action mention a requirement to have a free hand, so there's no such limitation for shielded arm.
Shield block generally requires a shield to be held rather than a free hand, though bucklers have some exceptions.
In this case there's no shield involved, so neither applies and the spell's target is free to raise their arm regardless of what's in their hand.