We report measurements of ac magnetic susceptibility $chi_{ac}$ and de Haas-van Alphen (dHvA) oscillations in KFe$_2$As$_2$ under high pressure up to 24.7 kbar. The pressure dependence of the superconducting transition temperature $T_c$ changes from negative to positive across $P_c sim 18$ kbar as previously reported. The ratio of the upper critical field to $T_c$, i.e, $B_{c2} / T_c$, is enhanced above $P_c$, and the shape of $chi_{ac}$ vs field curves qualitatively changes across $P_c$. DHvA oscillations smoothly evolve across $P_c$ and indicate no drastic change in the Fermi surface up to 24.7 kbar. Three dimensionality increases with pressure, while effective masses show decreasing trends. We suggest a crossover from a nodal to a full-gap $s$ wave as a possible explanation.
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