Optical nanoantennas can convert propagating light to local fields. The local-field responses can be engineered to exhibit nontrivial features in spatial, spectral and temporal domains, where local-field interferences play a key role. Here, we design nearly fully controllable local-field interferences in the nanogap of a nanoantenna, and experimentally demonstrate that in the nanogap, the spectral dispersion of the local-field response can exhibit tuneable Fano lineshapes with nearly vanishing Fano dips. A single quantum dot is precisely positioned in the nanogap to probe the spectral dispersions of the local-field responses. By controlling the excitation polarization, the asymmetry parameter q of the probed Fano lineshapes can be tuned from negative to positive values, and correspondingly, the Fano dips can be tuned across a broad spectral range. Notably, at the Fano dips, the local-field intensity is strongly suppressed by up to~50-fold, implying that the hot spot in the nanogap can be turned into a cold spot. The results may inspire diverse designs of local-field responses with novel spatial distributions, spectral dispersions and temporal dynamics, and expand the available toolbox for nanoscopy, spectroscopy, nano-optical quantum control and nanolithography.