Remember that fearful-avoidants want and desire contact but fear getting no response or feeling rejected.
When they pull away, “giving them space” only helps the part of them that needs to distance and that it a good thing, but it does not address the part of them that still wants connection.
In other words, if you give an avoidant “space”, they will appreciate the “space” but also feel rejected ad abandoned. If it sounds like someone conflicted, that is because it is.
When you give them more space then they needed, try to reopen the lines of communication, they’ll be cautious and sometimes even resistant to emotionally opening up because to a fearful-avoidant opening up means getting close and getting close ends in rejection.