An example of a state which yeilds an entangled state when you trace one qubit out, is the 3-qubit "W" state:
$$
\lvert W_3 \rangle = \tfrac{1}{\sqrt3} \Bigl( \lvert 100 \rangle + \lvert 010 \rangle + \lvert 001 \rangle \Bigr)
$$
Taking the outer product with itself, we obtain
$$
\lvert W_3 \rangle\!\langle W_3 \rvert = \tfrac{1}{3} \Bigl(
\!\begin{aligned}[t]&
\lvert 100 \rangle\!\langle 100 \rvert + \lvert 100 \rangle\!\langle 010 \rvert
+ \lvert 100 \rangle\!\langle 001 \rvert \phantom{\Big)}
\\&+
\lvert 010 \rangle\!\langle 100 \rvert + \lvert 010 \rangle\!\langle 010 \rvert
+ \lvert 010 \rangle\!\langle 001 \rvert
\\&+
\lvert 001 \rangle\!\langle 100 \rvert + \lvert 001 \rangle\!\langle 010 \rvert
+ \lvert 001 \rangle\!\langle 001 \rvert
\Bigr) \end{aligned}
$$
If we trace out the third qubit, we then obtain the state
$$
\begin{align}
\rho = \mathrm{tr}_3\Bigl(\lvert W_3 \rangle\!\langle W_3 \rvert\Bigr)
&= \tfrac{1}{3} \Bigl(
\lvert 10 \rangle\!\langle 10 \rvert + \lvert 10 \rangle\!\langle 01 \rvert +
\lvert 01 \rangle\!\langle 10 \rvert + \lvert 01 \rangle\!\langle 01 \rvert
+ \lvert 00 \rangle\!\langle 00 \rvert
\Bigr)
\\&= \tfrac{1}{3} \lvert 00 \rangle\!\langle 00 \rvert + \tfrac{2}{3} \lvert \Psi^+ \rangle\!\langle \Psi^+ \rvert,
\end{align}
$$
where in particular $\lvert \Psi^+ \rangle = \tfrac{1}{\sqrt 2}\bigl( \lvert 01 \rangle + \lvert 10 \rangle \bigr)$ is a Bell state.
Note that $\rho$ is a rank-2 operator with different eigenvalues: any decomposition of $\rho$ as a convex combination of other density operators, can only involve terms whose eigenvectors are supported on $\mathrm{span}\,\{ \lvert 00 \rangle, \lvert \Psi^+ \rangle \}$. Any mixed tensor product density operator $\rho_A \otimes \rho_B$ has an eigenbasis consisting of two or more product states; but the only product state contained in $\mathrm{span}\,\{ \lvert 00 \rangle, \lvert \Psi^+ \rangle \}$ is the state $\lvert 00 \rangle$ itself. It follows that $\rho$ cannot be decomposed as a convex combination of products of possibly-mixed states, and is entangled.
More generally, if for $n > 1$ we define $\lvert W_n \rangle$ as the analogous $n$-term uniform superposition of standard basis states with a single 1, we may describe it as
$$ \lvert W_n \rangle = \tfrac{\sqrt{n{-}1}}{\sqrt n} \lvert W_{n{-}1}\rangle\lvert0\rangle + \tfrac{1}{\sqrt n} \lvert00\cdots0\rangle\lvert1\rangle
$$
so that
$$ \mathrm{tr}_n\Bigl(\lvert W_n \rangle\!\langle W_n \rvert\Bigr) = \tfrac{n{-}1}{n} \lvert W_{n-1} \rangle\!\langle W_{n-1} \rvert + \tfrac{1}{n} \lvert 00\cdots0\rangle\!\langle 00\cdots 0\rvert
$$
which for larger values of $n$ is closer and closer to being a pure entangled state, while still being mixed for any finite $n$.