We have carried out high-field resistivity measurements up to 27,T in EuFe$_2$As$_2$ at $P$,=,2.5,GPa, a virtually optimal pressure for the $P$-induced superconductivity, where $T_mathrm{c}$,=,30,K. The $B_mathrm{c2}-T_mathrm{c}$ phase diagram has been constructed in a wide temperature range with a minimum temperature of 1.6 K ($approx 0.05 times T_mathrm{c}$), for both $B parallel ab$ ($B_mathrm{c2}^mathrm{ab}$) and $B parallel c$ ($B_mathrm{c2}^mathrm{c}$). The upper critical fields $B_mathrm{c2}^mathrm{ab}$(0) and $B_mathrm{c2}^mathrm{c}$(0), determined by the onset of resistive transitions, are 25 T and 22 T, respectively, which are significantly smaller than those of other Fe-based superconductors with similar values of $T_mathrm{c}$. The small $B_mathrm{c2}(0)$ values and the $B_mathrm{c2}(T)$ curves with positive curvature around 20 K can be explained by a multiple pair-breaking model that includes the exchange field due to the magnetic Eu$^{2+}$ moments. The anisotropy parameter, $Gamma=B_mathrm{c2}^{ab}/B_mathrm{c2}^{c}$, in EuFe$_2$As$_2$ at low temperatures is comparable to that of other 122 Fe-based systems.
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