The quartic \begin{equation} f = 3x^4 - 5x^2y^2 + 5y^4 + 20x^2y - 32y^3 + x^2 + 2y^2, \end{equation} which is even in $x$, has $5$ isolated local minima in \begin{equation} (0, 0), (\pm2, -1), (\pm2, 5). \end{equation} In order to prove the claim, one checks that the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ vanish and that the Hessian is positive definite in these points, where the latter is equivalent to the traces and the determinants being positive.
If one carries out the calculation by hand, then the symmetry $f(x,y)=f(-x,y)$ requires only to check $3$ points. Alternatively, the following Sage code verifies the example:
R.<x, y> = QQ[]
f = 3*x^4 - 5*x^2*y^2 + 5*y^4 + 20*x^2*y - 32*y^3 + x^2 + 2*y^2
fx, fy = f.derivative(x), f.derivative(y)
critical_points = ideal([fx, fy]).variety()
hesse = matrix(2, 2, [fx.derivative(x), fx.derivative(y),
fy.derivative(x), fy.derivative(y)])
minima = []
for cp in critical_points:
h = hesse.substitute(cp)
if h.trace() > 0 and h.det() > 0:
minima.append(cp)
print(len(minima) == 5)
Remark 1: As pointed out by Jap88 in the comments, previous examples are difficult to visualize. This holds true for this one too. The reason is that some of the saddles points tend to be very close to the minima. The red and blue lines are the curves given by $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$, and the green dots are the minima. As the first picture doesn't show well the situation around $(0,0)$, the second one zooms into this area:
Remark 2: Another rather short example is \begin{equation} f = x^4 - 3x^2y^2 + 3y^4 - 2x^2y + 3y^3 + x^2 - 2y^2. \end{equation} Here, the $5$ local minima are irrational, so the exact algebraic verification is a little more involved.
Remark 3: Conjecture 5.4 in this paper by Durfee et.al predicts that a quartic has at most $4$ isolated local minima. So this is a counterexample.
Remark 4: Theorem 3.1.6. of the paper Critical points of real polynomials ... by Shustin seems to claim that $5$ minima are possible. However, the "proof" given in Section 3.13 merely says "Assume that $d \le 4$. Then all the index distributions except [...] can be easily realized as it was explained above". I don't know what the author means by "above", as all the previous cases handle degrees $d\ge5$.