Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary.
If $M$ is an open subset of $\mathbb R^d$, we can consider the $\mathbb R$-Banach space $$C^1_b(M):=\left\{f:M\to\mathbb R\mid f\text{ is }C^1\text{-differentiable and }f\text{ and }{\rm D}f\text{ are bounded}\right\}$$ equipped with $$\left\|f\right\|_{C^1_b(M)}:=\max\left(\sup_{x\in M}\left|f(x)\right|,\sup_{x\in M}\left\|{\rm D}f(x)\right\|_{\mathfrak L(\mathbb R^d,\:\mathbb R)}\right)\;\;\;\text{for }f\in C^1_b(M).$$ Is there an analogous topological or (complete) normed $\mathbb R$-vector space of $C^1$-differentiable (bounded) functions in the general case?
I think the first problem is that the domain of the analogue to the Fréchet derivative, which is the pushforward $T_x(f):T_x\:M\to\mathbb R$, depends on $x\in M$. On the other hand, we could consider the map $$TM=\left\{(x,v):x\in M\text{ and }v\in T_x\:M\right\}\to\mathbb R\;,\;\;\;(x,v)\mapsto T_x(v)\tag1.$$
Consider a compact riemannian manifold $(M,g)$. Then $\mathcal{C}^1(M)$ has these natural norms : \begin{align} \|f\|_1 &= \sup_{x\in M}|f(x)| + \sup_{(x,v)\in TM, ||v||_g = 1}|\mathrm{d}f(x)v| \\ \| f \|_2 &= \sup\left(\sup_{x\in M}|f(x)|,\sup_{(x,v)\in TM, ||v||_g = 1}|\mathrm{d}f(x)v| \right) \end{align} You have to choose a way of measuring tangent vectors, so you have to choose a metric. This is why I required for $M$ to have such a metric $g$. This can also be done with a Finsler metric.