Probably this is well known to those who know it.

Got an argument and numerical support that over number fields elliptic curves in minimal models might have unbounded number of integral points, the number depending on the degree of the field.

Set $f(x)=x^3+ax+b$ and consider the curve $E: y^2=f(x)$.

Chose $x_1 \ldots x_n$ such that $f(x_n)$ is prime and work in $K=\mathbb{Q}[\sqrt{f(x_1)},\ldots\,\sqrt{f(x_n)}]$.

$E$ has the obvious $n$ points $(x_n,\sqrt{f(x_n)})$.

Experimentally for $f(x)=x^3-x+1$ over $\mathbb{Q}[\sqrt{7},\sqrt{61},\sqrt{211},\sqrt{337},\sqrt{991}]$ the five points are linearly independent according to sage so the rank is at least $5$.

Computing the absolute field is not efficient for me.

Over the rationals there is a conjecture relating the number of integral points to the rank, is there a similar conjecture for number fields?

Is there an example (with few primes) when in this construction the points are linearly dependent?

The same argument works for higher genus.