Is the tangent bundle of a submanifold a submanifold of the tangent bundle?

Question

If $N\subset M$ is an immersed submanifold of a smooth manifold $M$, is its tangent bundle $TN$ an immersed submanifold of $TM$?

In other words: If $\iota:N\to M$ is the immersion (smooth map with injective differential), is its differential $d\iota:TN\to TM $ also an immersion?

Intuitively, I think that the answer is yes but I have trouble proving it.