3. In the tetrahedron $P-ABC$, $BC=3$, $CA=4$, $AB=5$. If the dihedral angles between the three lateral faces and the base are $A-BC-P$ at $45^{\circ}$, $B-CA-P$ at $45^{\circ}$, and $C-AB-P$ at $45^{\circ}$, then the volume of the tetrahedron $P-ABC$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4
3. B.

Draw $P O \perp$ plane $A B C$ at $O, O D \perp B C$ at $D, O E \perp C A$ at $E, O F \perp A B$ at $F$, let $O P=h$. Then
$\angle P D O=45^{\circ}, \angle P E O=45^{\circ}, \angle P F O=45^{\circ}$.
Thus, $O D=h \cot 45^{\circ}=h, O E=h \cot 45^{\circ}=h$, $O F=h \cot 45^{\circ}=h$.
In $\triangle A B C$, $3 O D+4 O E+5 O F=2 S_{\triangle A B C}=12$,
Therefore, $3 h+4 h+5 h=12, h=1$.
Hence $V=\frac{1}{3} S h=\frac{1}{3} \times 6 \times 1=2$.