Are all strong bases alkalis?
Are all strong bases alkalis?


No, quaternary amine hydroxides such as tetramethylammonium hydroxide can also be strong bases.


  Can you not have a weak alkali?


According to "Acids and Bases" by Kristi Lew, beryllium hydroxide is the one and only exception to the "alkali bases are strong" rule.  

I would add that many of the pKb values of the alkali hydroxides are very close to zero.  So you should only think of the them as strong in the sense of strong means pKb near or less than zero.  If you think of strong as full dissociated even in a concentrated solution then they (NaOH for example) would not be strong by that definition. 

Also, for the alkaline earth metal hydroxides (which have two hydroxides per metal ion), it is only the first pKb that is strong.