Very drunk. In British slang, "pissed" means drunk (not angry, like in the US), so pissed to the gills is like being so drunk, you're SWIMMING (hence, the GILLS) in alcohol.
Last night I went down the pub and got pissed to the gills!
by CBizkit April 29, 2013
1 more definition
After being blown off by her her boyfriend, Cathy was pissed to the gills.
by cjsgirl February 05, 2010