  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.
