Yes, it is true that $R$ must be the field of real numbers.
As $R$ is an ordered field, it is naturally an extension $\mathbb{Q}\hookrightarrow R$. We can prove the following two properties, which characterize the reals among the ordered fields.
1) $\mathbb{Q}$ has no upper bound in $R$ (i.e., $R$ is Archimedean).
Proof: Call element $x$ of $R$ infinite if $\vert x\vert$ is an upper bound for $\mathbb{Q}$, and finite otherwise. Then we can define $f\colon R\to R$ by
$$
f(x)=\begin{cases}
\frac{x}{2}+\frac12\max(x,0)+(2+\max(x,0))^{-1},&\textrm{if }x\textrm{ is finite},\\\\
x/2,&\textrm{if }x\textrm{ is infinite}.
\end{cases}
$$
So,
- If $x,y$ are finite then they have an upper bound $a\ge0$ in $\mathbb{Q}$, and it can be seen that $\vert f(x)-f(y)\vert\le(1-(2+a)^{-2})\vert x-y\vert$.
- If $x,y$ are both infinite then $\vert f(x)-f(y)\vert=\frac12\vert x-y\vert$.
- If $x$ is infinite and $y$ is finite then $\vert f(x)-f(y)\vert\le \frac12\vert x\vert+\vert f(y)\vert\le\frac34\vert x-y\vert$.
In any case, if $\mathbb{Q}$ had an upper bound $\kappa\in R$ then we have $\vert f(x)-f(y)\vert\le (1-\kappa^{-1})\vert x-y\vert$ so that, by hypothesis, $f$ has a fixed point. But it can be seen that $f(x) > x$ for finite $x$ and $f(x)=\frac x2\not=x$ for infinite $x$. So, it doesn't have a fixed point, giving a contradiction.
2) Every Cauchy sequence $x_n$ in $R$ converges.
Proof: Passing to a subsequence1, it can be assumed that $x_n$ is monotonic, and replacing $x_n$ by $-x_n$ if necessary, we can suppose that it is increasing. If it is eventually constant then the result is immediate. Otherwise, by further passing to a subsequence2, we can suppose that $x_{n+2}-x_{n+1}\le\frac12(x_{n+1}-x_n)$ and that $x_{n+1}-x_n < 2^{-n-1}$. Then, $y_n=x_n+2^{-n}$ is a strictly decreasing sequence with $0\le y_n-x_n\le 2^{-n}$. Again, passing to a subsequence, it can be assumed that $y_{n+1}-y_{n+2}\le\frac12(y_n-y_{n+1})$.
We can define $f\colon R\to R$ linearly mapping $(-\infty,x_1]$ onto $(-\infty,x_2]$, $(x_n,x_{n+1}]$ onto $(x_{n+1},x_{n+2}]$, $[y_1,\infty)$ onto $[y_2,\infty)$, and $[y_{n+1},y_n)$ onto $[y_{n+2},y_{n+1})$ ($n\ge1$). This can be done such that $\vert f(x)-f(y)\vert\le\frac12\vert x-y\vert$ on each interval, in which case it does not have any fixed points in these intervals. Furthermore, if $x_n$ had no limit point, then the intervals cover3 $R$ and this defines $f$ everywhere. But, then, $\vert f(x)-f(y)\vert\le\frac12\vert x-y\vert$ for all $x,y\in R$ implying that $f$ has a fixed point, giving a contradiction.
I'll add a few more details that I passed over rather quickly above. A sequence $x_n$ is Cauchy if, for each $r > 0$ in $R$ then $\vert x_n - x_m\vert < r$ for large enough $m,n$. Any subsequence of a Cauchy sequence is itself Cauchy and tends to a limit $x$ if and only if the orginal sequence tends to $x$.
1 Any sequence in a linearly ordered set has a monotonic subsequence.
2 If $x_n$ is an increasing Cauchy sequence, which is not eventually constant, then it is possible to choose a subsequence $x_{n_k}$ as follows. Once $x_{n_k}$ has been chosen, then there is an $m > n_k$ such that $x_m \not= x_{n_k}$. As it is Cauchy, $n_{k+1}\ge m$ can be chosen such that $\vert x_r-x_s\vert < \min(2^{-k-2},(x_m-x_{n_k})/2)$ for all $r,s\ge n_{k+1}$. This ensures that $x_{n_{k+2}}-x_{n_{k+1}}$ is less than both $2^{-k-2}$ and $(x_{n_{k+1}}-x_{n_k})/2$ for all $k$.
3 If $z\in R$ was not in any of the intervals $(-\infty,x_1]$, $(x_n,x_{n+1}]$, $[y_1,\infty)$, $[y_{n+1},y_n)$ then $x_n < z < y_n$ for all $n$. So, $\vert z-x_n\vert\le y_n-x_n\le 2^{-n}$. Given any $r > 0$ in $R$, the fact that we have already shown $R$ to be Archimedean in (1) implies that $2^n > r^{-1}$ for large $n$. So, $\vert z - x_n\vert < r$ for large $n$, and $x_n\to z$.
