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 Abell 2744, 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 arcmin^2 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 ALMA 1.2-mm coverage, none are detected individually or in a stack. For the majority of the sample, SED 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 non-detections. While compact dusty star formation can not 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 by 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 = 10^43-10^46 erg/s, 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 AGN that are at least ten times more numerous than UV-luminous AGN at the same intrinsic bolometric luminosity.