Can a 2×6 beam span 11ft

A couple problems with the span calculation, IMHO (I am not a Licensed Professional Engineer):

Select structural (the lumber grade) is frequently impossible to come by at an ordinary lumberyard, will cost the earth if it can be had, and is even less available (AFAIK) in pressure treated (though you should not need that, as per comments.)

Number 2 is generally the grade you should design for unless you are sure you have a source for a better grade and can stand the price of it... You can get the same loading from 2x8 number 2 on 19.2" centers (5 per 8 feet - there's a symbol on most modern tape measures for that spacing) as your select structural on 24". Or you can get significantly more load (or span) from 2x10 number 2 on 24" centers

The position of this roof on the sidewall of a house (if it's "like the picture") may cause the buildup of a snowdrift on the roof - that needs to be allowed for if the 30 lb snow load is anything other than mythical (ie, I think that snow load is the code standard minimum, so it is often in code where snow is never expected....)

According to the table in the 2012 IRC code a exterior deck needs to handle a 60lb per sq ft (40 lb live + 20lb dead load). Most treated lumber that I am aware of is #2 southern yellow pine, using that criteria on the table in the code book, your joists can carry a 10'-9" span.

The biggest failure in deck construction has not been the posts or the joists, but the connection to the house. I believe code now requires through bolts into the band joist, or the joist of the house, or posts at the house is cool too.

The length of the post will dictate whether they need to be 4X4 or 4X6. Its a deflection (bowing) thing. IMO 4X4 for a 10' span is not overloading the post, a 16' span may...