Maximal ideals in $C^\infty(\mathbb{R})$
Solution 1:
I don't have an explicit example at hand, but there are other ideals than $\mathfrak m_p$. Note that $C_0^\infty$, the set of compactly supported smooth functions is an ideal, not contained in any $\mathfrak m_p$. By Zorn's lemma, every ideal is contained in some maximal ideal, so there must be others.
Solution 2:
No. For example, functions with compact support form an ideal.