A highly granular electromagnetic calorimeter with scintillator strip readout is being developed for future lepton collider experiments. A prototype of 21.5 $X_0$ depth and $180 times 180 $mm$^2$ transverse dimensions was constructed, consisting of 2160 individually read out $10 times 45 times 3$ mm$^3$ scintillator strips. This prototype was tested using electrons of 2--32 GeV at the Fermilab Test Beam Facility in 2009. Deviations from linear energy response were less than 1.1%, and the intrinsic energy resolution was determined to be $(12.5 pm 0.1 (mathrm{stat.}) pm0.4 (mathrm{syst.}))%/sqrt{E[mathrm{GeV}]}oplus (1.2 pm 0.1(mathrm{stat.})^{+0.6}_{-0.7}(mathrm{syst.}))%$, where the uncertainties correspond to statistical and systematic sources, respectively.