One crucial difference between Hilbert spaces and Banach spaces is that, in the latter, best approximations need not exist in the first place. Even if best approximations exist, the associated metric projection lacks most of the nice properties it has on Hilbert spaces.
Let's begin in the context of a Hilbert space $(H,(\cdot, \cdot))$. Consider a closed, convex subset $C \subset H$. Then for every $u \in H$ there exists a unique point $c=c(x) \in C$ that minimizes the distance
$$\operatorname{dist} (u,C) := \inf_{c \in C} \|u-c\|$$ i.e. $c$ is a the unique best approximation of $u$ by elements of $C$. The map
$$P_C \colon H \to C$$
that assosiates to each $u \in H$ its (unique) best approximation $c(x) \in C$ is called the metric projection and it is a generally nonlinear, nonexpansive (that is, $\|P(u) - P(v)\| \le \|u-v\|$) map that is characterized by the Kolmogorov criterion
$$ P_C(x)=y \iff y \in C \text{ and } (x-y,c-y) \leq 0 \text{ for every } c \in C .$$
In fact, $P_C$ is a linear bounded (with norm one) operator if and only if $C$ is a closed subspace. In that case $P_C$ is the orthogonal projection onto the subspace $C$ and is characterized by
$$ P_C(x) =y \iff x-y \perp C$$
When one works in the context of Banach spaces such a best approximation need not exist. One needs to consider Banach spaces that share some geometrical properties with Hilbert spaces. One such geometrical property is strict convexity. $X$ is called striclty convex if for any two distinct points of the unit sphere $x,y \in S_X = \{u \in X \colon \|u\| =1\}$ and any $\lambda \in (0,1),$
$$ \ \|\lambda x + (1-\lambda)y \| <1. $$
That is, $S_X$ contains no line segments.
A set $C \subset X$ such that, for every $x \in X$ there exists (at least one) best approximation of $x$ by elements of $C$ is called proximinal. If every $x$ admits a unique such best approximation then $C$ is called Chebyshev. It turns out ([M]) that
a) $X$ is reflexive if and only if every closed convex $C \subset X$ is proximinal.
b) $X$ is reflexive and stritly convex if and only if every closed convex $C \subset X$ is Chebyshev.
In the case where $C$ is a proximinal set, the metric projection is a multivalued map $P_C \colon X \to 2^C$ given by
$$P_C(x) = \{ y \in C \colon \operatorname{dist}(x,C) = \|x-y\|\}.$$
One would then need tools from multivalued map theory, which is an interesting topic on its own. Such tools would be notions of continuity, selections theorems, fixed point theorems etc.
In the case where $C$ is a Chebyshev set then the metric projection is single valued, and generally nonlinear and not continuous. It can be shown that it is continuous if $X$ is reflexive and has the Kadec-Klee property, i.e.
$$ x_n \rightharpoonup x \text{ (weakly) and } \|x_n\| \to \|x\| \implies x_n \to x $$
then $P_C$ is continuous. The most common example of such a space is a uniformly convex Banach space. It is generally hard to establish linearlity of the metric projection. An interesting result of Linderstrauss which states that if $X$ is a Banach space such that each of its closed subspaces admits a linear metric projection, then $X$ is (isomorphic to) a Hilbert space.
One last generalization that I am aware of, but this goes off topic, is considering sunny retractions that are nonexpansive. A sunny retraction is a map $P \colon C \to F$ where $C$ is closed convex and $F\subset C$ is closed such that
$$P(f)=f , \ \text{ for } f \in F$$
and
$$P( (1-λ)Px+λx) =Px,$$
whenever $\lambda \ge 0$ and $x,(1-λ)Px+λx \in C$. Notice that a metric projection is a sunny retractions. Recall that the (normalized) duality map is the multivalued map $J \colon X \to 2^{X^*}$ given by
$$ J(x) = \{x^* \in X^* \colon \langle x^*,x\rangle = \|x\| \ \|x^*\|, \ \|x^*\|=\|x\|\}$$
(which is nonempty due to Hahn Banach.) When $X$ is smooth, i.e. $J(x) = \{j(x)\}$ is a singleton for every $x \in X$, then the nonexpansive sunny retraction is characterized by
$$ \langle j(y-Px),x-Px \rangle \le 0, ~~~\forall x \in C,\forall y \in F.$$
Notice that this variational inequality is the Kolmogorov criterion when $X$ is a Hilbert space, since the duality map is the identity, i.e. $j(x)=x$.
An intersting method for calculating best approximations is the so-called Method of Alternating Projections (MAP for short). It is based on a theorem of Halperin [H] which states that if $M_1,\dots,M_k$ are closed subspaces of a Hilbert space $H$ and $M= \bigcap_{i=1}^k M_i$ then
$$ (P_{M_k} \dots P_{M_1})^n (x) \to P_M (x) , \ \forall x \in H,$$
i.e. the periodicic product of $P_{M_i}$ converges to $P_M$ is the strong operator topology.
This is helpful when the geometry of the problem makes it difficult to calculate $P_M(x)$ but it is much easier to calculate each $P_{M_i}(x)$. In fact, one can use quasi-periodic products of projections to approximate $P_M$. A sequence $\tau \colon \mathbb N \to \{1,\dots,k\}$ is called quasi-periodic if there exists $m\ge k$ such that for every $ n \in \mathbb N $,
$$ \{\tau(n) , \tau(n+1) , \dots , \tau (n+m-1)\} = \{1, \dots , k\} .$$
Sakai [S] showed that for any quasi-periodic sequence $\tau$,
$$ P_{τ(n)} \cdots P_{τ(1)}(x) \to P_M(x), \ \forall x \in H $$
One can find generalizations of this method for linear projections or even expansive sunny retractions in Banach spaces ([B-R]).
[B-R] Ronald E Bruck and Simeon Reich. Nonexpansive projections and resolvents
of accretive operators in banach spaces. Houston J. Math, 3(4):459–470,
1977.
[H] Israel Halperin. The product of projection operators. Acta Sci. Math.(Szeged),
23(1):96–99, 1962.
[M] Robert E Megginson. An introduction to Banach space theory, volume 183.
Springer Science & Business Media, 2012.
[S] Makoto Sakai. Strong convergence of infinite products of orthogonal projec-
tions in hilbert space. Applicable Analysis, 59(1-4):109–120, 1995.
