If a girl is trying to lock arms with a guy in a playful play while having a conversation, it may indicate that the girl is actually romantically interested in the guy.
Linking arms can be quite a romantic gesture, especially if it is followed up with a leaning of her head against the guy’s upper arm. This is often a sign that she feels both physically and emotionally safe with the guy.
In essence, she is letting her guard down and allowing him to be her protector.
If you have noticed that she isn’t this touchy-feely with other guys, there is a good chance that she is interested in you specifically and not just being friendly.