Using a sample of 92 UV continuum-selected, spectroscopically identified galaxies with z = 2.65, all of which have been imaged in the Lyα line with extremely deep narrow-band imaging, we examine galaxy Lyα emission profiles to very faint surface brightness limits. The galaxy sample is representative of spectroscopic samples of Lyman break galaxies (LBGs) at similar redshifts in terms of apparent magnitude, UV luminosity, inferred extinction, and star formation rate and was assembled without regard to Lyα emission properties. Approximately 45% (55%) of the galaxy spectra have Lyα appearing in net absorption (emission), with 20% satisfying commonly used criteria for the identification of "Lyα emitters" (LAEs; W 0 (Lyα) 20 Å). We use extremely deep stacks of rest-UV continuum and continuum-subtracted Lyα images to show that all sub-samples exhibit diffuse Lyα emission to radii of at least 10 (∼80 physical kpc). The characteristic exponential scale lengths for Lyα line emission exceed that of the λ 0 = 1220 Å UV continuum light by factors of ∼5-10. The surface brightness profiles of Lyα emission are strongly suppressed relative to the UV continuum light in the inner few kpc, by amounts that are tightly correlated with the galaxies' observed spectral morphology; however, all galaxy sub-subsamples, including that of galaxies for which Lyα appears in net absorption in the spectra, exhibit qualitatively similar diffuse Lyα emission halos. Accounting for the extended Lyα emission halos, which generally would not be detected in the slit spectra of individual objects or with typical narrow-band Lyα imaging, increases the total Lyα flux (and rest equivalent width W 0 (Lyα)) by an average factor of ∼5, and by a much larger factor for the 80% of LBGs not classified as LAEs. We argue that most, if not all, of the observed Lyα emission in the diffuse halos originates in the galaxy H ii regions but is scattered in our direction by H i gas in the galaxy's circum-galactic medium. The overall intensity of Lyα halos, but not the surface brightness distribution, is strongly correlated with the emission observed in the central ∼1 -more luminous halos are observed for galaxies with stronger central Lyα emission. We show that whether or not a galaxy is classified as a giant "Lyα blob" (LAB) depends sensitively on the Lyα surface brightness threshold reached by an observation. Accounting for diffuse Lyα halos, all LBGs would be LABs if surveys were sensitive to 10 times lower Lyα surface brightness thresholds; similarly, essentially all LBGs would qualify as LAEs.