A nonaqueous, extrudable composition includes at least one thermoplastic polymer in an amount of more than 20 wt % of the whole composition and tobacco. A smokeless tobacco product in the form of a sheet can be made by extruding or hot melt shaping a nonaqueous composition comprising at least one thermoplastic polymer and tobacco, the sheet being soluble in a user's mouth and resulting in sustained release of nicotine to the user. The sheet can be in a form that may be placed in the buccal cavity of, on the palate of or sublingually in the user, and have an average dissolution time of 5 to 50 minutes for delivering super bioavailable nicotine to the user.