Gateaux derivatives is a linear continuous operator In Clarke's book p.61, it is said that the Gâteaux derivatives $F'(x;v)$ at $x$ in the direction $v$ of $F:X\to Y$ ($X,Y$ being normed spaces) imply that $v\mapsto F'(x;v)$ is linear continuous.
But in the Wikipedia page (paragraph dedicated to Fréchet derivatives), first they say that in general $F'(x;v)$ may fail to be linear or continuous but in the Banach settings $F'(x;v)$ is linear.
Does Wikipedia miss the continuity property or the definition of the book is false in infinite dimensional settings ?
 A: It's not so much that Clarke's definition is "false," but that the definitions in this area are less standardized than you might expect.  The Wikipedia entry reflects this, implying in one place that the terms "Gateaux differential" and "Gateaux derivative" may two words for the same thing,

In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus . . .

while elsewhere explicitly noting that these might be different objects, depending upon the definitions one uses:

Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as Tikhomirov (2001), draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear).

I've added emphasis on "often" here, because as this language suggests, this choice of definitions is common but not universal.  Let's consider Clarke's definitions.
First, as Clarke defines the directional derivative (see page 20), if $X$ and $Y$ are normed spaces and $F$ is a function from $X$ to $Y$, then $F$ is said to have a directional derivative at $x \in X$ in the direction $v \in X$ if and only if the limit
$$
\lim_{t \to 0^{+}} \frac{F(x + tv) - F(x)}{t}
$$
exists (in the norm topology on $Y$).  When this is the case, Clarke denotes the value of the limit by $F'(x;v)$.
This definition is fairly standard, although it is already a departure from some sources, which use a two-sided limit instead of a one-sided limit here.  (This is what is currently done at https://en.wikipedia.org/wiki/Directional_derivative, for example.)
And this choice does make a difference in the notion of "directional derivative"; for example, if $f: \mathbb{R} \to \mathbb{R}$ is given by $f(x) = |x|$, then $f$ has a directional derivative at $0$ in the direction $1$ under Clarke's definition (because $\lim_{t \to 0^{+}} \frac{|t|}{t} = 1$), but not under the two-sided limit definition (because $\lim_{t \to 0} \frac{|t|}{t}$ does not exist).
Moving on to the notion of "Gateaux differentiable," for Clarke, the function $F$ is Gateaux differentiable at $x$ if $F'(x;v)$ exists for all $v \in X$, and if additionally there is norm-continuous linear map $L: X \to Y$ such that $F'(x;v) = L(v)$ holds for all $v \in X$.
Note, Clarke doesn't write this verbatim on his page 61: he writes "if there is $\Lambda \in L_C(X,Y)$ with $F'(x;v) = \langle \Lambda, v \rangle$ for all $v \in X$."  But $L_C(X,Y)$ is elsewhere defined (page 11) to be the set of norm-continuous, i.e., bounded, linear mappings from $X$ to $Y$.  And similarly, while Clarke does not appear to have actually defined the notation $\langle \Lambda, v \rangle$ for a general linear map $\Lambda: X \to Y$ and $v \in X$, he does define it in the case $Y = \mathbb{R}$ (see page 15), where it is sometimes called the 'dual pairing' or 'duality pairing,' and where it simply denotes $\Lambda(v)$.  We can infer from context that $\langle \Lambda, v \rangle$ is intended to mean $\Lambda(v)$ even if $Y$ is some normed space other than $\mathbb{R}$.  (I've switched from $\Lambda$ to $L$ because I am used to using upper case Roman letters for linear maps.)
So Clarke is explicitly requiring $L$ to be a continuous linear transformation.  Does this definition exclude some functions $F$ where there is some $x$ where $F'(x,v)$ is a linear function of $v$ that is not continuous?  Yes.
For one class of examples, pick an infinite-dimensional normed vector space $X$ and any discontinuous linear map $F: X \to \mathbb{R}$ (for examples of such $F$, see Clarke's page 12, or check out Discontinuous linear functional) and consider what is going on at $x = 0$.  For any $v \in X$ one has $\lim_{t \to 0^{+}} \frac{F(0+tv) - F(0)}{t} = \lim_{t \to 0^{+}} \frac{t F(v)}{t} = F(v)$, so that $F'(0;v)$ exists for all $v$, and is a linear function of $v$, but not a continuous one (because $F$ was chosen that way).
Such differences in definitions exist mainly because different authors have different priorities.  Some may want or need to consider the case of discontinuous linear maps, and others may not.  There's no one "right" set of definitions.  But if one is going to use a term like "Gateaux differentiable" in front of an unfamiliar audience, these examples show that it can be helpful to include the definition one is using.  It might matter!
