We identify an abundant population of extreme emission-line galaxies (EELGs) at redshift z {approx} 1.7 in the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey imaging from Hubble Space Telescope/Wide Field Camera 3 (HST/WFC3). Sixty-nine EELG candidates are selected by the large contribution of exceptionally bright emission lines to their near-infrared broadband magnitudes. Supported by spectroscopic confirmation of strong [O III] emission lines-with rest-frame equivalent widths {approx}1000 A-in the four candidates that have HST/WFC3 grism observations, we conclude that these objects are galaxies with {approx}10{sup 8} M{sub Sun} in stellar mass, undergoing an enormous starburst phase with M{sub *}/ M-dot{sub *} of only {approx}15 Myr. These bursts may cause outflows that are strong enough to produce cored dark matter profiles in low-mass galaxies. The individual star formation rates and the comoving number density (3.7 Multiplication-Sign 10{sup -4} Mpc{sup -3}) can produce in {approx}4 Gyr much of the stellar mass density that is presently contained in 10{sup 8}-10{sup 9} M{sub Sun} dwarf galaxies. Therefore, our observations provide a strong indication that many or even most of the stars in present-day dwarf galaxies formed in strong, short-lived bursts, mostly at z > 1.