I just construct an exact sequence $0\to M\to M\oplus N\to N\to0$ of $\mathbb{Z}$-modules that does not split, where $M=\mathbb{Z}$, $N=(\mathbb{Z}/2\mathbb{Z})^\\mathbb{N}$, and the map from $M$ to $M\oplus N$ maps $n$ to $(2n,0)$. Do you know other examples? You are free to consider other categories. In particular, are there any example that is in some sense "finite"? I guess such sequence must split for finitely generated modules over PID. Is this true?

(The above questions were mostly answered in the question that Martin points out. In particular, the answer to the last question is affirmative, even for commutative Noetherian rings instead of PID.)

More generally, is it true (e.g. for f.g. modules over PID) that any two exact sequence with the same terms

$0\to M\to P\to N\to0$

$0\to M\to P\to N\to0$

are equivalent, in the sense that there exist vertical isomorphisms which make the diagram commute? In other words, are there two extensions of $N$ by $M$ which are nonequivalent but both yield $P$? If not, can you give any counterexample? Thanks.

(It was noted in Ayoub's paper that this is not true in the category of finite groups, where he exhibited three finite groups that both fit in a split exact sequence and a non-split one.)