should I include offer letter from a different institution in my application for a faculty position I have a offer letter from a very well-known group in my field for a postdoc position. The proposed starting date would be early next year.
However, I am currently also applying for a faculty (assistant professor) position at a different institution.

Should I mention this offer in my application for the faculty position (on cv or cover letter) and/or include the official offer letter?

update:

there is another sublety for one application. One faculty position I consider requires experience abroad (i.e. not in the same country) of at least 1 year. I only have 6 months but the postdoc offer would be abroad giving me the 1 year total if I take it before starting the faculty position. Should I then mention the offer?

my field is health and medical sciences
I think that it would be unwise to include a copy of an offer in another application. 

But whether you inform them of the existence of an offer is a bit more subtle. I doubt that anyone will rush to hire you just because you have another offer. They will evaluate you on other things as usual. So, at best, mentioning the offer initially gets you nothing. 

However, later in the process if you need a decision and they are delaying making it, you can let them know that you have a deadline. Whether that helps or hurts is also subtle. Whether it is wise, or not depends on the nature of any relationship you have been able to build with them. In the absence of any relationship, it might hurt more than help. It is easy to just cross you off the list unless they are very interested and have few other interesting applicants.