Consider the irreducible polynomial $h(t)=t^n+c_{n-1}t^{n-1}+\cdots +c_1t+p \in \mathbb{Z}_p[t]$. Consider the ring of $p$-adic integers and maximal ideal $p \mathbb{Z}_p$.
I want to show all the coefficients $c_i, \ 1 \leq i \leq n-1$ lie in the maximal ideal $p \mathbb{Z}_p$.
I am thinking about Hensel's lemma.
If possible let, one of the coefficients $c_i$, say $c_k$ doesn't belong to $p \mathbb{Z}_p$, then reduction mod $p$ produces $$\bar{h}(t)=t^n+c_kt^k=t^k(t^{n-k}+c_k),$$ as a factorisation. Hence by Hensel's lemma the lift $h(t)$ of $\bar h(t)$ is reducible, which is a contradiction.
Thus all the coefficients $c_i$ lie in $p \mathbb{Z}_p$.
Am I correct ?
Your answer is ultimately correct, the only modification you need to make is that there could be arbitrarily many coefficients that aren't 0 mod p, in particular,
$$\bar h(t) = t^n + c_{i_1}t^{i_1} + \cdots + c_{i_k}t^{i_k} = t^{i_k}(t^{n-i_k} + c_{i_1}t^{i_1-i_k} + \cdots + c_{i_k})$$
Your factorization will still lift by Hensel's lemma like you describe because they satisfy the condition that they are relatively prime polynomials.