The James Webb Space Telescope (JWST) is revolutionizing our knowledge of z > 5 galaxies and their actively accreting black holes. Using the JWST Cycle 1 Treasury program Ultradeep NIRSpec and NIRCam Observations before the Epoch of Reionization (UNCOVER) in the lensing field A2744, we report the identification of a sample of little red dots at 3 < z
phot < 7 that likely contain highly reddened accreting supermassive black holes. Using a NIRCam-only selection to F444W < 27.7 mag, we find 26 sources over the ∼45 arcmin2 field that are blue in F115W − F200W ∼ 0 (or β
UV ∼ –2.0 for f
λ
∝ λ
β
), red in F200W − F444W = 1−4 (β
opt ∼ +2.0), and are dominated by a point-source-like central component. Of the 20 sources with deep Atacama Large Millimeter/submillimeter Array (ALMA) 1.2 mm coverage, none are detected individually or in a stack. For the majority of the sample, spectral energy distribution fits to the JWST+ALMA observations prefer models with hot dust rather than obscured star formation to reproduce the red NIRCam colors and ALMA 1.2 mm nondetections. While compact dusty star formation cannot be ruled out, the combination of extremely small sizes (〈r
e
〉 ≈ 50 pc after correction for magnification), red rest-frame optical slopes, and hot dust can be explained by reddened broad-line active galactic nuclei (AGNs). Our targets have faint M
1450 ≈ −14 to −18 mag but inferred bolometric luminosities of L
bol = 1043–1046 erg s−1, reflecting their obscured nature. If the candidates are confirmed as AGNs with upcoming UNCOVER spectroscopy, then we have found an abundant population of reddened luminous AGNs that are at least ten times more numerous than UV-luminous AGNs at the same intrinsic bolometric luminosity.