If I play a unit in my base, is my opponent able to play a hidden card?

No, your opponent cannot play a hidden card in response to you playing a unit to your base.

According to RiftJudge FAQ #7317:

• When you play a unit, it enters the chain as a pending item.
• When you finalize the unit, it is removed from the chain immediately.
• Units do not linger on the chain, and at no point does a player gain priority to respond to the unit itself.

Because playing a unit to your base does not create a triggered ability on the chain (unless that specific unit has a "When you play me" effect), there is no window for your opponent to play a reaction, including a hidden card.

If the unit you played has a "When you play me" effect, that ability would go on the chain after the unit enters play, and your opponent could then react to that triggered ability with a hidden card (provided the hidden card meets all targeting and timing restrictions). However, they cannot react to the act of playing the unit itself.