Resistivity and Hall effect measurements of EuFe$_2$As$_2$ up to 3.2,GPa indicate no divergence of quasiparticle effective mass at the pressure $P_mathrm{c}$ where the magnetic and structural transition disappears. This is corroborated by analysis of the temperature ($T$) dependence of the upper critical field. $T$-linear resistivity is observed at pressures slightly above $P_mathrm{c}$. The scattering rates for both electrons and holes are shown to be approximately $T$-linear. When a field is applied, a $T^2$ dependence is recovered, indicating that the origin of the $T$-linear dependence is spin fluctuations.