The Briton, looking to achieve a rare Tour-Vuelta double, lost almost half a minute to the big guns in the 18.7-km climb to the Alto de Capileira at the end of a 191.1-km ride from Jodar.
Lindeman, of the Lotto-Jumbo team, won the stage ahead of Belarussian Ilia Koshevoy (Lampre).
Italian Fabio Aru (Astana), one of the top contenders, accelerated away from the group of favourites after realising Froome was being dropped.
Aru finished 29 seconds behind Lindeman while the group of favourites, featuring overall leader Esteban Chaves of Colombia (Orica-GreenEdge), crossed the line 36 seconds off the pace.
Team Sky rider Froome cracked with under two kilometres left and ended up 17th, 1:03 behind Lindeman who made his winning move inside the last 500 metres.
Chaves leads Dutchman Tom Dumoulin (Giant-Alpecin) by 10 seconds overall, and Ireland's Dan Martin, who will join Etixx-Quick Step from Cannondale-Garmin next season, by 33 seconds.
Saturday's eighth stage is a 182.5-km ride from Puebla de Don Fadrique to Murcia.
(Reporting by Julien Pretot; Editing by Tony Jimenez)
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