Can a wireless headphone (bluetooth) contain malware?

Can a wireless (bluetooth) headphone store data? Also, if it can, then can someone put malware in it? I assume that the device must necessarily be able to store some sort of data, since it has buttons for skip and pause songs and buttons to change the volume. So it must contain some software to do this built inside it.

Is there anyway to check (if it's really possible that it can contain malware) if it's infected or not?

Thank you.