Although the Toffees have a game in hand of those around them in the race, they are seven points behind fourth-placed West Ham with just eight games to play.
Earlier in the day, WBA kept their slim hopes of survival alive by recording a thumping 3-0 win over Southampton, who also missed a late penalty.
Sam Allardyce's side are now eight points short of safety with seven games to play.
At the summit of the table, meanwhile, Manchester United kept their very slim title hopes alive with an impressive 3-1 comeback victory over Tottenham in London on Sunday.
Goals from Fred, Edinson Cavani and Mason Greenwood overturned a first-half Son strike to close the gap on leaders Manchester City to 11 points.
Man Utd still have a game in hand and, while a title triumph is highly unlikely, they have at least pushed back City's title celebrations after their local rivals lost at home to Leeds on Saturday.
Lingard now has nine goals in eight games for the Hammers since moving on loan in January from Man Utd.
Arsenal have almost no chance of finishing in the top four but they did earn a morale-boosting 3-0 win at rock-bottom Sheffield United. Alexandre Lacazette (two) and Gabriel Martinelli grabbed the goals.
Steve Bruce's men now move six points clear of Fulham in the final relegation spot and they still have a game in hand to come.
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