Can I mount my hangboard on a wooden ceiling beam?
Structural engineer here. 

The beam is almost surely not stressed to it's limit. I'm not sure where you are at, that is, if you are in a snow area, but roofs are designed for loads that don't happen all the time such as snow, water, construction, etc.

Also, wood as a material has the ability to resist higher short term loads.

For a beam that size, I would have no problem with what you propose. Because, if that beam  needs to be that size due to loads then the additional 180 lb live load is relatively small. My main concern, as stated in another answer, is HOW you attach it.

Generally you'll be fine with a few screws say of the typical size #8 (length not important) anywhere. If you need larger, stay away from the lower 1/3 of the beam where it is in tension.