Sarcasm

Can sarcasm be intended to tell a joke? I'd say yes. I flippantly and jokingly told a lady recently that I wore my name badge for fun and she took the huge huge huge, huge as a house offense to it because it was sarcasm.
The first entry for sarcasm in dictionary.com is:
–noun

1.	harsh or bitter derision or irony.

2.	a sharply ironical taunt; sneering or cutting remark

The lady I was talking to stated the Greek meaning of "tearing of the flesh" and asked for an apology. I hadn't said it in an acidic manner, but lighthearted one if I remember correctly. I thought at first she was joking, but, she was deadly serious, so I of course apologized. I hadn't meant to be intentionally really mean. At the same time after a full minute talking to me and asking me about this and that she goes "oh by the way, do you work here?". She honest to goodness in all that time had not noticed the shinny name badge I was wearing and said as much. I dunno what to think.