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I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

If $p$ is odd, let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, and take $d=5$ if $p=2$, so that $K(\sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above, if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because each degenerare fibre is a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156v1 [math.AG]), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

If $p$ is odd, let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, and take $d=5$ if $p=2$, so that $K(\sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above, if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because each degenerare fibre is a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group $A_0(X)_0$ of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59, available at Numdam) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

Addendum (in response to a question in an email I received). One can show moreover that no smooth projective surface $Y$ over $\mathbf{Q}_p$ which is $\mathbf{Q}_p$-birational to $X$ can have good reduction. This follows from the facts recalled above and the theorem of Colliot-Thélène and Coray (which can be found in Fulton's Intersection theory) : $A_0(Y)_0$ is isomorphic to $A_0(X)_0$.

I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

If $p$ is odd, let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, and take $d=5$ if $p=2$, so that $K(\sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because the degenerare fibres are a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156v1 [math.AG]), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

If $p$ is odd, let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, and take $d=5$ if $p=2$, so that $K(\sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above, if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because each degenerare fibre is a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156v1 [math.AG]), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

Assume that $p$ is odd and let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, so that $K(\sqrt{d})|K$ is the unramified quadratic extension. (The method can be made to work for the prime $2$ by replacing $y^2-d$ by $y^2-y-1$ in what follows, because $\mathbb{Q}_2(\sqrt{5})|\mathbb{Q}_2$ is the unramified quadratic extension.)

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because the degenerare fibres are a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156v1 [math.AG]), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

If $p$ is odd, let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, and take $d=5$ if $p=2$, so that $K(\sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because the degenerare fibres are a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156v1 [math.AG]), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.